Stochastic Integration in Lp Spacesantoni/media/diploma thesis antoni.pdf · Developing the theory of stochastic integration in these spaces will be the subject of this thesis. Due

Stochastic Integrationin Lp Spaces

Diploma Thesis

of

Markus Antoni

at the Institute for Analysis

of the Department of Mathematics

Reviewer: Prof. Dr. Lutz Weis

Second reviewer: Prof. Dr. Roland Schnaubelt

KIT – University of the State of Baden-Wuerttemberg and National Laboratory of the Helmholtz Association www.kit.edu

Contents

Introduction 1

1 Random Series 5

1.1 The Kahane Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 The Karhunen-Loeve Representation Theorem . . . . . . . . . . . . . . 11

1.3 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Martingale Inequalities 27

2.1 Martingale Difference Sequences . . . . . . . . . . . . . . . . . . . . . . 27

2.2 The Strong Doob and Strong Burkholder-Gundy Inequality . . . . . . . 29

2.2.1 The Reduction Process . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.2 The UMD Property . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2.3 The Strong Doob Inequality . . . . . . . . . . . . . . . . . . . . . 41

2.2.4 The Strong Burkholder-Gundy Inequality . . . . . . . . . . . . . 45

2.3 Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Stochastic Integration 49

3.1 The Wiener Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 The Ito Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.3 The Integral Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.4 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4 An Application to Stochastic Evolution Equations 87

4.1 Ito Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.2 The Ito Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.3 An Infinite Dimensional Version of the Geometric Brownian Motion . . 108

A Appendix 113

A.1 Integration in Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . 113

A.1.1 Measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

A.1.2 The Bochner Integral . . . . . . . . . . . . . . . . . . . . . . . . 116

A.2 Gaussian Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . 119

A.3 Conditional Expectations and Martingales . . . . . . . . . . . . . . . . . 122

Bibliography 131

Introduction

In 1923, a stochastic integral∫ 1

0f dβ was first introduced by Norbert Wiener in

[16] for functions f ∈ L2([0, 1]) with respect to a Brownian motion β. In this

case the following isometry holds

E∣∣∣ ∫ 1

0

f dβ∣∣∣2 = ‖f‖2

L2([0,1]).

21 years later, Kiyoshi Ito first presented a stochastic integral in [6] for adapted

processes f ∈ L2([0, 1]× Ω), where he showed the famous Ito isometry

E∣∣∣ ∫ 1

0

f dβ∣∣∣2 = ‖f‖2

L2([0,1]×Ω).

From this theory the stochastic calculus arose, greatly influenced by Ito’s for-

mula. Today, these results are widely applied in various fields, especially in

financial mathematics.

Based on this theory, it is now more or less easy to verify that these stochastic

integrals and their isometries can be generalized to the Hilbert space valued

setting. This is due to the fact that the norm comes from an inner product.

For functions with values in an arbitrary Banach space stochastic integration is

more difficult. In [11] by Jan van Neerven and Lutz Weis and in [10] by Jan van

Neerven, Marc Veraar, and Lutz Weis, the authors studied stochastic integrability

of operator-valued functions Φ: [0, T ] × Ω → B(H,E). Here, H is a separable

Hilbert space and E is a UMD space. An example of the latter one are the Lp

spaces for 1 < p <∞. In general, UMD spaces are those Banach spaces E which

satisfy

E∥∥∥ N∑n=1

εndn

∥∥∥qE.q,E E

∥∥∥ N∑n=1

dn

∥∥∥qE

for any sequence (εn)Nn=1 ⊆ [−1, 1], each martingale difference sequence (dn)Nn=1,

and some (and hence all) 1 < q <∞.

With this in mind, one may ask whether we could get new results or less elabo-

rate proofs of the existent results if we restrict ourselves to Lp spaces by taking

2 Introduction

advantage of the structure we have there. Developing the theory of stochastic

integration in these spaces will be the subject of this thesis. Due to the fact

that, in what follows, the number p is reserved for another purpose, we will from

now on replace the p with an r and speak of Lr spaces. More precisely, we will

consider the space Lr(U,Σ, µ) for 1 ≤ r <∞ (or 1 < r <∞, respectively), where

(U,Σ, µ) is a σ-finite measure space and Σ is countably generated. Note that in

this case Lr(U,Σ, µ) is separable.

The results of this thesis are based on various results from stochastic analysis

and martingale theory. Among other things, we will give a less elaborate proof of

the Kahane inequalities for Gaussian and Rademacher sums, and we will show a

representation theorem for Gaussian random variables. This will be the topic of

Chapter 1, which will be completed by an introduction of the Brownian motion.

In Chapter 2, we take a closer look on martingale difference sequences and the

UMD property. The highlight of this chapter may be a new (stronger) version

of Doob’s maximal inequality, which states that for 1 < p, r < ∞ and suitable

Lr(U,Σ, µ)-valued martingales (Mn)Nn=1 we have

E∥∥ N

maxn=1|Mn|

∥∥pr.p,r E ‖MN‖pr .

This will then also lead to a new version of the Burkholder-Gundy inequality

stating that

E∥∥ N

maxn=1|Mn|

∥∥prhp,r E

∥∥∥( N∑n=1

|dn|2) 1

2∥∥∥pr

for 1 < p, r < ∞. Here, (dn)Nn=1 is the martingale difference sequence of the

Lr(U,Σ, µ)-valued martingale (Mn)Nn=1.

Having laid these foundations, we will continue with the theory of stochastic

integration in Chapter 3. We will first construct a stochastic integral for functions

f : [0, T ] → Lr(U,Σ, µ), where 1 ≤ r < ∞, and we will see that those integrals

belong to the family of Gaussian random variables. In the next step, we present

the stochastic integral for appropriate processes f : [0, T ]×Ω→ Lr(U,Σ, µ), now

assuming that 1 < r < ∞. Based on that, we will study the integral process(∫ t0f dβ

)t∈[0,T ]

, which happens to be a martingale. Applying the results from

Chapter 2 will therefore lead to a Burkholder-Gundy inequality for stochastic

integrals, i.e.,

E∥∥∥ supt∈[0,T ]

∣∣∣ ∫ t

0

f dβ∣∣∣ ∥∥∥p

rhp,r E

∥∥∥(∫ T

0

|f |2 dt) 1

2∥∥∥pr

for 1 < p <∞. In the last part of this chapter, we will finally use a localization

argument to augment the class of integrands significantly.

Introduction 3

In the final chapter we consider Ito processes, i.e., processes of the form

X(t) = x0 +

∫ t

0

f ds+∞∑n=1

∫ t

0

bn dβn, t ∈ [0, T ],

for suitable functions f and bn, n ∈ N. Here, (βn)∞n=1 is a sequence of independent

Brownian motions. At this point, we will show an interesting connection to the

operator-valued stochastic integral we mentioned above. We also prove the Ito

formula, which will subsequently be used to solve an abstract stochastic evolution

equation.

Finally, in the appendix we provide an introduction to the theory of integration in

Banach spaces, present some facts about Gaussian random variables, and show

some results regarding vector-valued conditional expectations and martingales

that are used in this thesis. Especially for readers who have not collected some

experience in these fields we recommend to read the appendix first.

Notations. In this thesis, (U,Σ, µ) will always be a σ-finite measure space with

countably generated σ-algebra Σ. For a number 1 ≤ r ≤ ∞, we define Lr(U) :=

Lr(U,Σ, µ) as a Banach space over R, and the Holder conjugate of r by r′ := rr−1

(with 1′ :=∞ and∞′ := 1). The norm of this space will be abbreviated by ‖ · ‖r.Moreover, for 1 ≤ r < ∞, we identify the duality space Lr(U)∗ with the space

Lr′(U) since every continuous functional T ∈ Lr(U)∗ is given by a multiplication

operator with unique kernel g ∈ Lr′(U), i.e., Tg(f) := T (f) =∫Ufg dµ. Thus,

by

〈f, g〉 := 〈f, Tg〉 = Tg(f) =

∫U

fg dµ

we denote the duality pairing of the elements f ∈ Lr(U) and g ∈ Lr′(U). If

a ≤ C(q) b for nonnegative numbers a and b and a constant C(q) > 0 depending

only on the variable q, we write a .q b. Additionally, we write a hq b if a .q band b .q a. Finally, for real numbers x and y, we define x ∨ y := maxx, y and

x ∧ y := minx, y.

Acknowledgements. First of all, I would like to express my deepest appreci-

ation to my advisor Lutz Weis for providing outstanding support and guidance

at every stage of this thesis. Further, I would like to thank him, Maria Girardi,

and Holly Watson for the kind hospitality during my stay at the University of

South Carolina, where parts of this thesis were made. Many thanks go to Roland

Schnaubelt for the pleasure to enjoy four years of excellent teaching. Finally, I

would like to express my gratitude to my fellow students, especially Christiane

Gorgen, Jorg Bauerle, and Alexander Weber, for five years of great assistance

throughout my studies.

Chapter 1

Random Series

In this chapter we collect some estimates for sums of independent random vari-

ables. More precisely, we concentrate on sums of the form∑N

n=1 γnxn and∑Nn=1 rnxn, where the γn are R-valued Gaussian variables, rn are Rademacher

variables, and the xn are elements of Lr(U). In section 1.2 we will then see that

such sums are the basic modules for general Lr(U)-valued Gaussian random vari-

ables, and in Chapter 3 we again run across these sums since stochastic integrals

of Lr(U)-valued step functions are of this form. In section 1.3 we investigate

the Brownian motion process and use results of section 1.2 to prove certain path

properties of Brownian motions.

1.1 The Kahane Inequality

Let (Ω,A ,P) be a probability space. We call a random variable r : Ω→ −1,+1a Rademacher variable (or Bernoulli variable) if

P(r = 1) = P(r = −1) =1

2.

A random variable γ : Ω→ R is said to be Gaussian if its distribution has density

fγ : R→ R, fγ(t) =1√

2πσ2exp(−1

2t2

σ2

)with respect to the Lebesgue measure on R for some σ ≥ 0. Note that a Gaussian

variable is always centered, i.e., the mean value of a Gaussian variable is 0. If

σ = 1, γ is called a standard Gaussian variable.

For the rest of this section, (rn)∞n=1 and (γn)∞n=1 will always denote a sequence

of independent Rademacher variables and a sequence of independent standard

Gaussian variables, respectively.

6 Random Series

Lemma 1.1. For all 1 ≤ p <∞ and all finite sequences α1, . . . , αN ∈ R, we

have (E∣∣∣ N∑n=1

αnγn

∣∣∣p) 1p

= c(p)

( N∑n=1

|αn|2) 1

2

,

where c(p) =√

2π−12pΓ(p+1

2

) 1p and Γ is the Gamma function defined by

Γ: (0,∞)→ R, Γ(x) =

∫ ∞0

tx−1e−t dt.

Proof. First we compute for r > −1

1√2π

∫ ∞−∞|t|r exp

(− t2

2

)dt =

√2π

∫ ∞0

tr exp(− t2

2

)dt

=√

2π

2r−12

∫ ∞0

sr−12 exp(−s) ds

=√

2π

2r−12 Γ(r+1

2

),

where we used the substitution s = t2

2in the second equality. If αn = 0 for each

1 ≤ n ≤ N , there is nothing to proof. So we can assume that there exists at

least one αk with αk 6= 0. Next, by the independence of the Gaussian sequence,

we remark that the linear combination∑N

n=1 αnγn is again a Gaussian variable

with variance∑N

n=1 α2n. Using the foregoing computation, we obtain

E∣∣∣ N∑n=1

αnγn

∣∣∣p =1√

2π∑N

n=1 α2n

∫ ∞−∞|t|p exp

(−1

2t2∑N

n=1 α2n

)dt

=1√2π

( N∑n=1

α2n

) p2∫ ∞−∞|s|p exp

(− s2

2

)ds

=√

2π

2p−12 Γ(p+1

2

)( N∑n=1

α2n

) p2

,

applying the substitution s = t(∑N

n=1 α2n

)− 12 in the second equality.

If the Gaussian sequence in Lemma 1.1 is replaced by a Rademacher sequence,

we do not get equality but at least some sort of equivalence. For the proof of this

estimate, we first need the following proposition.

1.1 The Kahane Inequality 7

Proposition 1.2. Let f : Ω → R be a measurable function. Then for all

1 ≤ p <∞ we have∫Ω

|f |p dP =

∫ ∞0

pλp−1P(|f | ≥ λ

)dλ.

Proof. With Fubini’s theorem we get∫ ∞0

pλp−1P(|f | ≥ λ

)dλ =

∫ ∞0

pλp−1

∫Ω

1|f |≥λ dP dλ =

∫Ω

∫ |f |0

pλp−1 dλ dP

=

∫Ω

|f |p dP.

Theorem 1.3 (Khintchine inequality). For all 1 ≤ p < ∞ and all finite

sequences α1, . . . , αN ∈ R, we have(E∣∣∣ N∑n=1

αnrn

∣∣∣p) 1p

hp

( N∑n=1

|αn|2) 1

2

.

Proof. (1) We first consider the upper bound. In view of

2nn! ≤ (2n)! for all n ∈ N0,

we get for all x ∈ R

1

2

(exp(x) + exp(−x)

)=

1

2

( ∞∑n=0

xn

n!+∞∑n=0

(−x)n

n!

)=∞∑n=0

x2n

(2n)!

≤∞∑n=0

(12x2)n

n!= exp

(12x2).

(4)

Now let t > 0 be fixed and define Ω+ :=t∑N

n=1 αnrn ≥ 0

. Then

E exp

(∣∣∣ t N∑n=1

αnrn

∣∣∣) =

∫Ω+

exp(t

N∑n=1

αnrn

)dP +

∫ΩC+

exp(−t

N∑n=1

αnrn

)dP

≤ E exp(t

N∑n=1

αnrn

)+ E exp

(−t

N∑n=1

αnrn

)= E

( N∏n=1

exp(tαnrn)

)+ E

( N∏n=1

exp(−tαnrn)

)= (?).

8 Random Series

Next, using the stochastic independence of the Rademacher sequence and (4),

we obtain

(?) =N∏n=1

E exp(tαnrn) +N∏n=1

E exp(−tαnrn)

=N∏n=1

1

2

(exp(tαn) + exp(−tαn)

)+

N∏n=1

1

2

(exp(−tαn) + exp(tαn)

)≤ 2

N∏n=1

exp(t2

2α2n

)= 2 exp

(t2

2

N∑n=1

α2n

).

Therefore, by Chebyshev’s inequality, we deduce that

P(∣∣∣ N∑

n=1

αnrn

∣∣∣ ≥ λ

)≤ exp(−tλ)E exp

(∣∣∣ t N∑n=1

αnrn

∣∣∣)

≤ 2 exp(−tλ+ t2

2

N∑n=1

α2n

)for any λ > 0. Taking t := λ∑N

n=1 α2n

leads to

P(∣∣∣ N∑

n=1

αnrn

∣∣∣ ≥ λ

)≤ 2 exp

(−1

2

λ2∑Nn=1 α

2n

).

Finally, Proposition 1.2 yields

E∣∣∣ N∑n=1

αnrn

∣∣∣p =

∫ ∞0

pλp−1P(∣∣∣ N∑

n=1

αnrn

∣∣∣ ≥ λ

)dλ

≤ 2

∫ ∞0

pλp−1 exp

(−1

2

λ2∑Nn=1 α

2n

)dλ

= p 2p2

( N∑n=1

α2n

) p2∫ ∞

0

sp2−1 exp(−s) ds

= p 2p2 Γ(p2

)( N∑n=1

α2n

) p2

,

where we used the substitution s = 12

λ2∑Nn=1 α

2n.

(2) First, notice that for each n = 1, . . . , N we have

Ern = 0 and Er2n = 1.

1.1 The Kahane Inequality 9

Therefore, using the stochastic independence of the Rademacher sequence, we

obtain

E∣∣∣ N∑n=1

αnrn

∣∣∣2 =N∑n=1

N∑m=1

E(αnαmrnrm)

=N∑n=1

E(αnrn)2 +N∑n=1

N∑m=1,m 6=n

αnαm Ern Erm

=N∑n=1

α2n.

By Holder’s inequality and step (1) (with p = 4) we get

N∑n=1

α2n = E

∣∣∣ N∑n=1

αnrn

∣∣∣2 = E∣∣∣ N∑n=1

αnrn

∣∣∣ 23 ∣∣∣ N∑n=1

αnrn

∣∣∣ 43≤(E∣∣∣ N∑n=1

αnrn

∣∣∣) 23(E∣∣∣ N∑n=1

αnrn

∣∣∣4) 13

≤(E∣∣∣ N∑n=1

αnrn

∣∣∣) 23

1613

( N∑n=1

α2n

) 23

,

and thus we obtain( N∑n=1

α2n

) 12

≤ 4 E∣∣∣ N∑n=1

αnrn

∣∣∣ ≤ 4

(E∣∣∣ N∑n=1

αnrn

∣∣∣p) 1p

for all 1 ≤ p <∞.

In what follows, we assume that 1 ≤ r < ∞ is fixed. Next, we consider finite

sequences x1, . . . , xN ∈ Lr(U) instead of coefficients α1, . . . , αN ∈ R.

Theorem 1.4 (Kahane inequalities). Let 1 ≤ p <∞ and set q := p ∨ r.Then we have for all x1, . . . , xN ∈ Lr(U)(

E∥∥∥ N∑n=1

γnxn

∥∥∥pr

) 1p

hq

∥∥∥( N∑n=1

|xn|2) 1

2∥∥∥r

and (E∥∥∥ N∑n=1

rnxn

∥∥∥pr

) 1p

hq

∥∥∥( N∑n=1

|xn|2) 1

2∥∥∥r.

10 Random Series

Proof. Let (ξn)Nn=1 ∈

(γn)Nn=1, (rn)Nn=1

. Due to Lemma 1.1 or Theorem 1.3,

respectively, we have(E∣∣∣ N∑n=1

ξnxn(u)∣∣∣p) 1

p

hp

( N∑n=1

|xn(u)|2) 1

2

for each fixed u ∈ U . Thus, we obtain

∥∥∥(E∣∣∣ N∑n=1

ξnxn

∣∣∣p) 1p∥∥∥rhp

∥∥∥( N∑n=1

|xn|2) 1

2∥∥∥r.

(1) For p ≥ r, using Minkowski’s integral inequality, we have(E∥∥∥ N∑n=1

ξnxn

∥∥∥pr

) 1p

=

(∫Ω

(∫U

∣∣∣ N∑n=1

ξnxn

∣∣∣r dµ

) pr

dP) 1

p

≤(∫

U

(∫Ω

∣∣∣ N∑n=1

ξnxn

∣∣∣p dP) r

p

dµ

) 1r

=∥∥∥(E∣∣∣ N∑

n=1

ξnxn

∣∣∣p) 1p∥∥∥rhp

∥∥∥( N∑n=1

|xn|2) 1

2∥∥∥r.

For p < r, Holder’s inequality and the foregoing estimate lead to(E∥∥∥ N∑n=1

ξnxn

∥∥∥pr

) 1p

≤(E∥∥∥ N∑n=1

ξnxn

∥∥∥rr

) 1r

.r∥∥∥( N∑

n=1

|xn|2) 1

2∥∥∥r.

(2) For the converse inequality, we apply Proposition A.10 and again Holder’s

inequality to get

∥∥∥( N∑n=1

|xn|2) 1

2∥∥∥r.∥∥∥E∣∣∣ N∑

n=1

ξnxn

∣∣∣ ∥∥∥r≤ E

∥∥∥ N∑n=1

ξnxn

∥∥∥r≤(E∥∥∥ N∑n=1

ξnxn

∥∥∥pr

) 1p

for all 1 ≤ p <∞.

Remark 1.5. As an immediate consequence of Theorem 1.4, we obtain the fol-

lowing estimate (E∥∥∥ N∑n=1

ξ1nxn

∥∥∥pr

) 1p

hp,q,r

(E∥∥∥ N∑n=1

ξ2nxn

∥∥∥qr

) 1q

for all 1 ≤ p, q < ∞ and every sequence x1, . . . , xN ∈ Lr(U), where (ξ1n)Nn=1,

(ξ2n)Nn=1 ∈

(γn)Nn=1, (rn)Nn=1

can be chosen arbitrarily.

1.2 The Karhunen-Loeve Representation Theorem 11

The next corollary shows that, in the situation of a convergent Gaussian or

Rademacher sum, we even have unconditional convergence.

Corollary 1.6 (Contraction principle). Let (εn)Nn=1 ⊆ [−1, 1] and 1 ≤p <∞. Then we have for all x1, . . . , xN ∈ Lr(U)(

E∥∥∥ N∑n=1

εnγnxn

∥∥∥pr

) 1p

.p,r

(E∥∥∥ N∑n=1

γnxn

∥∥∥pr

) 1p

and (E∥∥∥ N∑n=1

εnrnxn

∥∥∥pr

) 1p

.p,r

(E∥∥∥ N∑n=1

rnxn

∥∥∥pr

) 1p

.

1.2 The Karhunen-Loeve Representation Theo-

rem

In this section we will discuss some properties of Lr(U)-valued Gaussian ran-

dom variables, which are defined as a function X : Ω → Lr(U) on a probability

space (Ω,A ,P) such that the R-valued random variable 〈X, g〉 is Gaussian for

all g ∈ Lr′(U), where 1 ≤ r < ∞ is again fixed. The main result will be a rep-

resentation theorem, which states that every Gaussian random variable can be

represented as a (convergent) sum of the form∑∞

n=1 γnxn. Note that some aux-

iliary results regarding Gaussian random variables are outlined in the Appendix.

First we consider the fundamental integrability result for Gaussian random vari-

ables due to Fernique.

Theorem 1.7 (Fernique). Let X be an Lr(U)-valued Gaussian random

variable. Then there exists a constant β > 0 such that

E exp(β‖X‖2

r

)<∞.

Proof. (1) Let Y be an independent copy of X, and let t ≥ s > 0 be fixed.

Then, by Proposition A.15,

V :=X + Y√

2and W :=

X − Y√2

12 Random Series

are independent and have the same distribution as X and Y . Moreover, observe

that the set (x, y) ∈ R2

+ : |x− y| ≤ s√

2, x+ y > t√

2

is a subset of (x, y) ∈ R2

+ : x >t− s√

2, y >

t− s√2

.

Therefore, we obtain

P(‖X‖r ≤ s

)· P(‖Y ‖r > t

)= P

(∥∥∥ X − Y√2

∥∥∥r≤ s

)· P(∥∥∥ X + Y√

2

∥∥∥r> t

)≤ P

(∣∣∣ ‖X‖r − ‖Y ‖r√2

∣∣∣ ≤ s,‖X‖r + ‖Y ‖r√

2> t

)≤ P

(‖X‖r >

t− s√2, ‖Y ‖r >

t− s√2

)= P

(‖X‖r >

t− s√2

)· P(‖Y ‖r >

t− s√2

).

Since X and Y have the same distribution, this leads to

P(‖X‖r ≤ s

)· P(‖X‖r > t

)≤ P

(‖X‖r >

t− s√2

)2

.

(2) Choose s0 > 0 such that P(‖X‖r ≤ s0

)≥ 2

3, and define

t0 := s0 and tn := s0 +√

2 tn−1 for n ≥ 1.

By induction we then have tn = s0

√2n+1−1√2−1

, and since√

2 − 1 > 14

√2, we get

tn ≤ s0

√2n+4

. For n ∈ N, we define

αn :=P(‖X‖r > tn

)P(‖X‖r ≤ s0

) .By construction we have α0 ≤ (1− 2

3)/2

3= 1

2, and by step (1) (with s = s0 and

t = tn+1) we obtain

αn+1 =P(‖X‖r > tn+1

)P(‖X‖r ≤ s0

) ≤ P(‖X‖r >

tn+1−s0√2

)2

P(‖X‖r ≤ s0

)2 =P(‖X‖r > tn

)2

P(‖X‖r ≤ s0

)2 = α2n.

It follows that

P(‖X‖r > tn

)= αnP

(‖X‖r ≤ s0

)≤ α2n

0 ≤ 2−2n .


Using these estimates, we obtain

E exp(β‖X‖2

r

)≤ P

(‖X‖r ≤ t0

)exp(βt20)

+∞∑n=0

P(tn < ‖X‖r ≤ tn+1

)exp(βt2n+1)

≤ exp(βs20) +

∞∑n=0

2−2n exp(βs20 2n+5)

= exp(βs20) +

∞∑n=0

exp(2n(− log 2 + 32βs2

0))

=: C.

Now take β < 132s20

log 2 and apply Cauchy’s root test to see that C <∞.

Remark 1.8. Let 1 ≤ p < ∞, and take α := 1 + log(dpe!). By an elementary

computation we have xp ≤ eαx2

for every x ≥ 0. As a consequence of Theorem

1.7, we thus get √βα

p

E‖X‖pr ≤ E exp(β ‖X‖2

r

)<∞,

and hence

E‖X‖pr <∞,

where the bounding constant does not depend on r.

As a simple corollary to Fernique’s theorem, we get the following result.

Corollary 1.9. If X is an Lr(U)-valued Gaussian random variable, then we

have EX = 0.

Proof. By Remark 1.8 we have E‖X‖r <∞, and therefore

〈EX, g〉 = E〈X, g〉 = 0 for all g ∈ Lr′(U).

The claim now follows by the Hahn-Banach theorem.

Example 1.10. Let X be an Lr(U)-valued random variable of the form X =∑∞n=1 γnxn, where (γn)∞n=1 is a sequence of independent Gaussian variables and

(xn)∞n=1 is a (finite or infinite) sequence in Lr(U). Obviously, XN :=∑N

n=1 γnxn is

a Gaussian random variable for each N ∈ N, and the sequence (XN)∞N=1 satisfies

limN→∞

〈XN , g〉 = 〈X, g〉 almost surely for all g ∈ Lr′(U).

Hence, by Proposition A.17, X is a Gaussian random variable.

14 Random Series

This example shows that every (convergent) Gaussian sum is a Gaussian ran-

dom variable. Next, we turn our attention to the question if every Lr(U)-valued

Gaussian random variable can be represented as a Gaussian sum of the form∑∞n=1 γnxn. The next example shows that this is true in the case Lr(U) = RN .

Example 1.11. The theory of RN -valued Gaussian random variables and the

notions we will use in this example can be found in many introducing works

about stochastic calculus, for example in [9].

Assume that X is an RN -valued Gaussian random variable with positive-definite

covariance matrix Cov(X) ∈ RN×N . Since Cov(X) is symmetric, there exists an

orthonormal basis v1, . . . , vN ⊆ RN of eigenvectors of Cov(X) with correspond-

ing eigenvalues λ1, . . . , λN > 0. Set

A := (v1| . . . |vN) · diag(√λ1, . . . ,

√λN).

Then A is invertible and Cov(X) = AAT . Now set Y := A−1X. Then Y is a

Gaussian random variable with Cov(Y ) = Id, which means that the components

of Y are independent standard Gaussian variables. Write Y = (γ1, . . . , γN) for a

Gaussian sequence (γn)Nn=1. Then we obtain the following representation of X

X = AY =N∑n=1

γn√λnvn.

The next result shows that the desired estimate is also true in arbitrary Lr spaces.

Theorem 1.12 (Karhunen-Loeve). Let X be an Lr(U)-valued random

variable. Then X is Gaussian if and only if it has the form

X =∞∑n=1

γnxn,

where convergence holds almost surely and in Lp(Ω;Lr(U)) for all 1 ≤ p <∞.

Here, (γn)∞n=1 is a sequence of independent standard Gaussian variables and

(xn)∞n=1 ⊆ Lr(U). If this is the case, we further have

(E‖X‖pr

) 1p hr,p

∥∥∥( ∞∑n=1

|xn|2) 1

2 ∥∥∥r<∞

for one (or all) 1 ≤ p <∞.


For the proof of this theorem we need the following convergence result.

Proposition 1.13. Fix 1 ≤ p < ∞. Let (Xn)∞n=1 be a sequence of in-

dependent, integrable, and centered Lr(U)-valued random variables. Put

SN :=∑N

n=1Xn, and let S ∈ Lp(Ω;Lr(U)) satisfying limN→∞ 〈SN , g〉 = 〈S, g〉in L1(Ω) for all g ∈ Lr′(U). Then we have

limN→∞

SN = S almost surely and in Lp(Ω;Lr(U)).

Proof. Since limN→∞ 〈SN , g〉 = 〈S, g〉 in L1(Ω) for all g ∈ Lr′(U), we have

limN→∞

∣∣∣ ∫B

〈SN , g〉 − 〈S, g〉 dP∣∣∣ ≤ lim

N→∞

∫Ω

∣∣〈SN , g〉 − 〈S, g〉∣∣ dP = 0

for every B ∈ A . Next, define FN := σ(X1, . . . , XN) ⊆ A . Then every SN is

FN -measurable, and Xn is independent of FM whenever n > M . Hence, using

(A.2),(A.5), and the fact that EXn = 0 for each n, we get for N > M

E[SN |FM ] =M∑n=1

Xn +N∑

n=M+1

EXn =M∑n=1

Xn = SM .

So, (SN)∞N=1 is a martingale with respect to (FN)∞N=1. Now fix a K ∈ N. Then,

by the foregoing convergence result, we obtain⟨∫B

S dP, g⟩

=

∫B

〈S, g〉 dP = limN→∞

∫B

〈SN , g〉 dP

= limN→∞

⟨∫B

E[SN |FK ] dP, g⟩

=⟨∫

B

SK dP, g⟩

for all g ∈ Lr′(U) and every B ∈ FK . By the Hahn-Banach theorem and

Theorem A.21 we almost surely have SK = E[S|FK ] for all K ∈ N. Since

S ∈ Lp(Ω;Lr(U)), the martingale convergence theorem yield

limN→∞

SN = limN→∞

E[S|FN ] = E[S|F∞]

almost surely and in Lp(Ω;Lr(U)). Moreover, we have for all g ∈ Lr′(U)⟨E[S|F∞], g

⟩= lim

N→∞〈SN , g〉 = 〈S, g〉 almost surely,

and hence E[S|F∞] = S almost surely by Corollary A.8.

16 Random Series

Proof (of Theorem 1.12). If X =∑∞

n=1 γnxn, then Example 1.10 implies that

X is Gaussian.

For the converse direction, we define

GX :=〈X, g〉 : g ∈ Lr′(U)

L2(Ω)⊆ L2(Ω).

Then(GX , ‖ · ‖L2(Ω)

)is a (separable) Hilbert space and, by Proposition A.17,

every random variable in GX is Gaussian. Choose an orthonormal basis (γn)∞n=1

of GX , and observe that the orthogonality of (γn)∞n=1 in GX implies stochastic

independence (cf. Proposition A.16). Next, consider the linear mapping

ΦX : GX → Lr(U), ΦX(γ) = EγX,

which is well-defined and bounded. In fact, by applying Holder’s inequality and

Fernique’s theorem, we obtain

‖ΦX(γ)‖r ≤ E|γ|‖X‖r ≤(E|γ|2

) 12(E‖X‖2

r

) 12 .X

(E|γ|2

) 12 .

Now define xn := ΦX(γn), and set XN :=∑N

n=1 γnxn for N ∈ N. We then get for

all g ∈ Lr′(U)

limN→∞

〈XN , g〉 = limN→∞

N∑n=1

γn〈ΦX(γn), g〉 = limN→∞

N∑n=1

γnEγn〈X, g〉 = 〈X, g〉 in GX .

By Fernique’s theorem we have X ∈ Lp(Ω;Lr(U)) for each 1 ≤ p <∞. Therefore,

Proposition 1.13 implies that

limN→∞

XN = X almost surely and in Lp(Ω;Lr(U))

for all 1 ≤ p <∞. Finally, by the Kahane inequality and Fernique’s theorem,

(E‖X‖pr

) 1p =

(E∥∥∥ ∞∑n=1

γnxn

∥∥∥pr

) 1p

hp,r

∥∥∥( ∞∑n=1

|xn|2) 1

2∥∥∥r<∞.

As a consequence of Theorem 1.12, we get the following corollary.

Corollary 1.14 (Kahane inequality for Gaussian random variables).

Let X be an Lr(U)-valued Gaussian random variable. Then we have for all

1 ≤ p, q <∞ (E‖X‖pr

) 1p hp,q,r

(E‖X‖qr

) 1q .


As an application of this corollary, we want to show that if a sequence of Gaus-

sian random variables converges in probability, then it converges in Lp for all

1 ≤ p <∞. For this purpose we need the following lemma.

Lemma 1.15 (Paley-Zygmund inequality). Let ξ be a non-negative ran-

dom variable satisfying

0 < Eξ2 ≤ c(Eξ)2<∞

for some c > 0. Then, for all 0 < r < 1 we have

P(ξ > rEξ) ≥ (1− r)2

c.

Proof. Since ξ is non-negative, we have

(1− r)Eξ = E(ξ − rEξ) ≤ E(1ξ>rEξ(ξ − rEξ)

)≤ E(1ξ>rEξξ).

Therefore, using the Cauchy-Schwarz inequality, we get

(1− r)2(Eξ)2 ≤(E(1ξ>rEξξ)

)2 ≤ E1ξ>rEξ Eξ2.

Finally, dividing both sides by Eξ2, we obtain

P(ξ > rEξ) ≥ (1− r)2 (Eξ)2

Eξ2≥ (1− r)2

c.

Theorem 1.16. For a sequence (Xn)∞n=1 of Lr(U)-valued Gaussian random

variables the following assertions are equivalent:

(1) the sequence (Xn)∞n=1 converges in probability to a random variable X;

(2) for some 1 ≤ p < ∞, the sequence (Xn)∞n=1 converges in Lp(Ω;Lr(U))

to a random variable X;

(3) for all 1 ≤ p < ∞, the sequence (Xn)∞n=1 converges in Lp(Ω;Lr(U)) to

a random variable X.

In this case, the random variable X is Gaussian.

Proof. Since the applications (3) ⇒ (2) ⇒ (1) are clear, we only have to prove

that limn→∞Xn = X in probability implies limn→∞Xn = X in Lp(Ω;Lr(U)) for

all 1 ≤ p <∞. Note that X is Gaussian by Proposition A.17.

18 Random Series

(1) Fix 1 ≤ q <∞. By Fernique’s theorem we have E‖Xn‖qr <∞ for all n ∈ N.

We now want to show that even

supn∈N

E‖Xn‖qr <∞.

By Corollary 1.14 (with p = 4 and q = 2), there exists a constant c := c(r, 4, 2) >

0 such that

0 < E‖Xn‖4r ≤ c4

(E‖Xn‖2

r

)2for all n ∈ N.

Therefore, we can apply the Paley-Zygmund inequality and obtain

P(‖Xn‖2

r >12E‖Xn‖2

r

)≥ 1

4 c4. (4)

Now let ε > 0 be arbitrary. Since limn→∞Xn = X in probability, we can find for

any t > 0 an index N ∈ N such that for all n ≥ N we have

P(‖Xn‖2

r > t)≤ P

(‖X‖r > 1

2

√t)

+ P(‖Xn −X‖r > 1

2

√t)

≤ P(‖X‖r > 1

2

√t)

+ ε,

where we used that‖X‖r ≤ 1

2

√t, ‖Xn −X‖r ≤ 1

2

√t⊆‖Xn‖2

r ≤ t.

Hence, for t0 > 0 large enough, we find an index N0 > 0 such that for all n ≥ N0

P(‖Xn‖2

r > t0)< 2ε.

Next, assume that there exists a subsequence such that limk→∞ E‖Xnk‖2r = ∞.

Then, for all sufficiently large k, we obtain

P(‖Xnk‖2

r >12E‖Xnk‖2

r

)≤ P

(‖Xnk‖2

r > t0)< 2ε,

which contradicts (4). Therefore, supn∈N E ‖Xn‖2r < ∞. Using Corollary 1.14

again, we finally get

supn∈N

E‖Xn‖qr .r,q,2 supn∈N

E‖Xn‖2r <∞.

(2) Fix 1 ≤ p < q < ∞. By step (1) and Fernique’s theorem, there exists a

constant C > 0 such that

supn∈N

(E‖Xn −X‖qr

) 1q ≤ sup

n∈N

(E‖Xn‖qr

) 1q +

(E‖X‖qr

) 1q ≤ C.

1.3 Brownian Motion 19

Let ε > 0 be fixed. Using Holders inequality (with 1p

= 1q

+ 1s), we obtain

E‖Xn −X‖pr = E(1‖Xn−X‖r≤ε‖Xn −X‖pr

)+ E

(1‖Xn−X‖r>ε‖Xn −X‖pr

)≤ εp + E

(1‖Xn−X‖r>ε‖Xn −X‖pr

)≤ εp +

(E1‖Xn−X‖r>ε

) ps(E‖Xn −X‖qr

) pq

≤ εp + Cp P(‖Xn −X‖r > ε

) ps .

Since limn→∞Xn = X in probability, it follows that

limn→∞

E‖Xn −X‖pr < εp.

This being true for all ε > 0, we get limn→∞ E‖Xn −X‖pr = 0.

1.3 Brownian Motion

In this section we want to give an introduction to the Brownian motion, which

plays an important role in the theory of stochastic integration. Let E be an arbi-

trary Banach space. Then an E-valued stochastic process, indexed by a set I, is

a family of E-valued random variables(X(i)

)i∈I defined on a probability space

(Ω,A ,P).

Definition 1.17. An R-valued process(β(t)

)t∈[0,T ]

is called Brownian mo-

tion (or Wiener process) if the following properties hold:

(1) β(0) = 0 almost surely;

(2) for all 0 ≤ s ≤ t ≤ T , β(t)− β(s) is Gaussian with variance t− s;

(3) for all 0 ≤ s ≤ t ≤ T , β(t)− β(s) is independent of β(r) : 0 ≤ r ≤ s.

Remark 1.18. The definition of a Brownian motion often include a fourth prop-

erty, which states that:

(4) the function t 7→ β(t) is almost surely continuous.

As we will see in Remark 1.23, we can drop this property since every Brownian

motion with the properties (1)–(3) has a version which even has Holder continuous

trajectories.

20 Random Series

Remark 1.19. Every Brownian motion(β(t)

)t∈[0,T ]

is a martingale with respect

to the filtration (F βt )t∈[0,T ] defined by

F βt := σ

(β(s) : s ∈ [0, T ]

).

To see this, we first observe that(β(t)

)t∈[0,T ]

is adapted to (F βt )t∈[0,T ] and the

random variables β(t) are integrable. So it remains to show that E[β(t)|F βs ] =

β(s) almost surely for all 0 ≤ s ≤ t ≤ T . Since β(s) is F βs -measurable and

β(t)−β(s) is independent of F βs (this follows from the definition of the Brownian

motion), (A.2) and (A.5) lead to

E[β(t)|F βs ] = E[β(s)|F β

s ] + E[β(t)− β(s)|F βs ]

= β(s) + E(β(t)− β(s)

)= β(s).

We next want to investigate the existence of Brownian motions. Therefore, let

(γn)∞n=1 be a sequence of independent standard Gaussian variables, and let (gn)∞n=1

be an orthonormal basis in L2([0, T ]). For n ∈ N we then define

Gn : [0, T ]→ R, Gn(t) =

∫ t

0

gn ds.

For 0 < γ < 1, we denote by Cγ([0, T ]) the space of all γ-Holder continuous

functions, which is a Banach space when endowed with the norm

‖f‖Cγ := sups,t∈[0,T ]

|f(t)− f(s)||t− s|γ

+ |f(0)|.

Theorem 1.20. The series∑∞

n=1 Gnγn converges almost surely in Cγ([0, T ])

and in Lp(Ω;Cγ([0, T ])) for γ < 12. Additionally,

(∑∞n=1Gn(t)γn

)t∈[0,T ]

is a

Brownian motion.

To prove Theorem 1.20, we first have to show the next two propositions.

Proposition 1.21. Define the Beta function by

B : (0,∞)× (0,∞)→ R, B(x, y) =

∫ 1

0

tx−1(1− t)y−1 dt.

Then we have

Γ(x)Γ(y) = Γ(x+ y)B(x, y),

where Γ is the Gamma function from Lemma 1.1.


Proof. Define φ : (0,∞) × (0, 1) → (0,∞) × (0,∞) by φ(u, v) =(uv, u(1 − v)

).

Then |detφ′(u, v)| = u > 0, and φ is surjective, which means that φ is a diffeo-

morphism. Applying the change of variables formula and Fubini’s theorem, we

obtain

Γ(x)Γ(y) =

(∫ ∞0

sx−1e−s ds

)(∫ ∞0

ty−1e−t dt

)=

∫(0,∞)×(0,∞)

sx−1ty−1e−x−y d(s, t)

=

∫(0,∞)×(0,1)

ux−1vx−1uy−1(1− v)y−1e−uu d(u, v)

=

(∫ ∞0

ux+y−1e−u du

)(∫ 1

0

vx−1 (1− v)y−1 dv

)= Γ(x+ y)B(x, y).

Proposition 1.22. For 0 < α ≤ 1 and suitable functions f : [0, T ] → R we

define the operator Dα by

(Dαf)(t) := [kα ∗ f ](t) =

∫ t

0

kα(t− s)f(s) ds, 0 ≤ t ≤ T,

where kα(s) := 1Γ(α)

sα−1. Then we have:

(1) DαDβ = Dα+β for α, β > 0 with 0 < α + β ≤ 1;

(2) Dα : L2([0, T ])→ Lr([0, T ]) is continuous for 12< α ≤ 1 and 1 ≤ r <∞.

Further let Let 0 ≤ γ < 12. Then

(3) Dβ : Lr([0, T ]) → Cγ([0, T ]) is continuous for γ + 1r< β ≤ 1 and

2 < r <∞.

Proof. (1) Let α, β > 0 with 0 < α + β ≤ 1. Then, with Proposition 1.21,

(kα ∗ kβ) (t) =

∫ t

0

kα(t− s)kβ(s) ds =1

Γ(α)Γ(β)

∫ t

0

(t− s)α−1sβ−1 ds

=1

Γ(α)Γ(β)

∫ t

0

tα−1(

1− s

t

)α−1

sβ−1 ds

=1

Γ(α)Γ(β)tα+β−1

∫ 1

0

uα−1(1− u)β−1 du

=B(α, β)

Γ(α)Γ(β)tα+β−1 =

1

Γ(α + β)tα+β−1 = kα+β(t),

22 Random Series

where we used the substitution u = 1 − st

in the fourth equation. Using this

together with the associativity of the convolution, we get

DαDβf =(kα ∗ (kβ ∗ f)

)=((kα ∗ kβ) ∗ f

)= (kα+β ∗ f) = Dα+βf

for appropriate functions f .

(2) Now let 12< α ≤ 1 and 1 ≤ r < ∞. Note that Dα is linear and, by the

Cauchy-Schwarz inequality,

‖Dαf‖rr =

∫ T

0

∣∣(Dαf)(t)∣∣r dt

≤∫ T

0

(∫ t

0

∣∣kα(t− s)f(s)∣∣ ds)r dt

≤∫ T

0

(∫ t

0

kα(t− s)2 ds

) r2(∫ t

0

f(s)2 ds

) r2

dt

=

∫ T

0

(1

Γ(α)√

2α− 1tα−

12

)r(∫ t

0

f(s)2 ds

) r2

dt

≤ c1(α, r, T )r(∫ T

0

f(s)2 ds

) r2

= c1(α, r, T )r‖f‖r2,

where c1(α, r, T ) = 1Γ(α)

√2α−1

Tα−12

+ 1r . Thus, Dα : L2([0, T ]) → Lr([0, T ]) is con-

tinuous.

(3) Now fix 0 < γ < 12, and let 2 < r < ∞ and γ + 1

r< β ≤ 1. Then, for all

0 ≤ s ≤ t ≤ T ,∣∣(Dβf)(t)− (Dβf)(s)∣∣

=1

Γ(β)

∣∣∣∫ t

0

(t− u)β−1f(u) du−∫ s

0

(s− u)β−1f(u) du∣∣∣

≤ 1

Γ(β)

∫ s

0

∣∣((t− u)β−1 − (s− u)β−1)f(u)

∣∣ du+

1

Γ(β)

∫ t

s

∣∣(t− u)β−1f(u)∣∣ du.

Next, we estimate each summand separately. Observe that for all x, y ≥ 0 and

all q ≥ 1 we have

|x− y|q ≤ |xq − yq|.

In fact, for x > y ≥ 0 we have

(x− y)q = xq(1− y

x

)q ≤ xq(1− y

x

)= xq − xq y

x≤ xq − xq

(yx

)q= xq − yq,


which gives the desired estimate. Using this together with Holder’s inequality,

we obtain∫ s

0

∣∣((t− u)β−1 − (s− u)β−1)f(u)

∣∣ du≤(∫ s

0

∣∣(t− u)β−1 − (s− u)β−1∣∣r′ du) 1

r′

‖f‖r

≤(∫ s

0

∣∣(t− u)(β−1)r′ − (s− u)(β−1)r′∣∣ du) 1

r′

‖f‖r

=

(∫ s

0

(s− u)(β−1)r′ − (t− u)(β−1)r′ du

) 1r′

‖f‖r

=

(1

(β − 1)r′ + 1

((t− s)(β−1)r′+1 + s(β−1)r′+1 − t(β−1)r′+1

)) 1r′

‖f‖r

≤(

1

(β − 1)r′ + 1(t− s)(β−1)r′+1

) 1r′

‖f‖r.

And similarly,∫ t

s

∣∣(t− u)β−1f(u)∣∣ du ≤ (∫ t

s

∣∣(t− u)β−1∣∣r′ du) 1

r′

‖f‖r

=

(∫ t

s

(t− u)(β−1)r′ du

) 1r′

‖f‖r

=

(1

(β − 1)r′ + 1(t− s)(β−1)r′+1

) 1r′

‖f‖r.

Therefore,∣∣(Dβf)(t)− (Dβf)(s)∣∣ ≤ 2

Γ(β)

(1

(β − 1)r′ + 1

) 1r′

(t− s)β−1+ 1r′ ‖f‖r

=2

Γ(β)

(1

(β − 1)r′ + 1

) 1r′

(t− s)β−1r ‖f‖r.

Using this estimate and observing that β − 1r− γ > 0, we obtain

‖Dβf‖Cγ = sups,t∈[0,T ]

|(Dβf)(t)− (Dβf)(s)||t− s|γ

+ |(Dβf)(0)|

≤ 2

Γ(β)

(1

(β − 1)r′ + 1

) 1r′

sups,t∈[0,T ]

|t− s|β−1r−γ‖f‖r

≤ c2(β, γ, r, T )‖f‖r,

with c2(β, γ, r, T ) := 2Γ(β)

(1

(β−1)r′+1

) 1r′ T β−

1r−γ. This concludes the proof.

24 Random Series

Proof (of Theorem 1.20). (1) Let 2 < r <∞ such that γ+ 1r< 1

2, and choose

12< α < 1 and γ+ 1

r< β < 1

2such that α+β = 1. Then, by Proposition 1.22 (1),

we have Gn = DβDαgn, and by Proposition 1.22 (2) and (3) it suffices to prove

that∑∞

n=1(Dαgn)γn converges almost surely in Lr([0, T ]) and in Lp(Ω;Lr([0, T ])).

By the Bessel inequality we obtain for all t ∈ [0, T ]( N∑n=1

(Dαgn)(t)2

) 12

≤( ∞∑n=1

(∫ T

0

1(0,t)(s)kα(t− s)gn(s) ds

)2) 12

≤∥∥1(0,t)kα(t− ·)

∥∥2

=1

Γ(α)

(∫ t

0

s2(α−1) ds

) 12

=1

Γ(α)√

2α− 1tα−

12 ,

which is integrable. Moreover, this estimate shows that∑∞

n=1(Dαgn)(t)2 con-

verges absolutely for each t ∈ [0, T ]. By the Kahane inequality and the dominated

convergence theorem, we thus obtain

limN,M→∞

(E∥∥∥ M∑n=N

(Dαgn)γn

∥∥∥rr

) 1r

hr limN,M→∞

∥∥∥( M∑n=N

(Dαgn)2) 1

2∥∥∥r

= 0,

which implies that∑∞

n=1(Dαgn)γn converges in Lr(Ω;Lr([0, T ])). Combining

Theorem 1.12 and Theorem 1.16, we infer that∑∞

n=1(Dαgn)γn converges almost

surely in Lr([0, T ]) and in Lp(Ω;Lr([0, T ])) for all 1 ≤ p <∞.

(2) By step (1),(X(t)

)t∈[0,T ]

defined by X(t) :=∑∞

n=1 Gn(t)γn is an R-valued

stochastic process, which satisfies X(0) = 0 almost surely. Fix 0 ≤ s ≤ t ≤ T .

Using the dominated convergence theorem, the L2-orthogonality of the Gaussian

sequence (γn)∞n=1, and the Parseval identity, we obtain

E(X(s)X(t)

)=∞∑n=1

∞∑m=1

Gn(s)Gm(t)E(γnγm) =∞∑n=1

Gn(s)Gn(t)

=∞∑n=1

(∫ T

0

1(0,s)(u)gn(u) du

)(∫ T

0

1(0,t)(u)gn(u) du

)=

∫ T

0

1(0,s)(u)1(0,t)(u) du = s.

By Theorem 1.12, X(t)−X(s) is Gaussian with variance

E(X(t)−X(s)

)2= EX(t)2 − 2E

(X(s)X(t)

)+ EX(s)2 = t− 2s+ s = t− s.

Next, we want to show that X(t) − X(s) is independent of(X(r1), . . . , X(rN)

)whenever 0 ≤ r1 ≤ . . . ≤ rN ≤ s. For this it suffices to prove that the random


variables X(r1), X(r2)−X(r1), . . . , X(rN)−X(rN−1), X(t)−X(s) are indepen-

dent. By construction and Theorem 1.12,

N∑n=1

an(X(rn)−X(rn−1)

)+ aN+1

(X(t)−X(s)

)is Gaussian for all sequences (an)N+1

n=1 (where we put r0 := 0). By Proposition A.16

we therefore obtain the desired result if we show their orthogonality in L2(Ω).

As above, we compute

E[(X(t)−X(s)

)(X(rn)−X(rn−1)

)]= EX(t)X(rn)− EX(t)X(rn−1)− EX(s)X(rn) + EX(s)X(rn−1)

= rn − rn−1 − rn + rn−1 = 0

for all n = 1, . . . , N . Similarly, we get for all 1 ≤ m < n ≤ N

E[(X(rn)−X(rn−1)

)(X(rm)−X(rm−1)

)]= 0.

Thus, by definition,(X(t)

)t∈[0,T ]

is a Brownian motion.

Remark 1.23. Assume that(β(t)

)t∈[0,T ]

is a Brownian motion. By Fubini’s

theorem, we have

E∫ T

0

β(t)2 dt =

∫ T

0

Eβ(t)2 dt =

∫ T

0

t dt =1

2T 2 <∞.

Therefore, β ∈ L2([0, T ]) almost surely. Moreover,

Cγ([0, T ]) :=f ∈ L2([0, T ]) : ∃ f ∈ f with f ∈ Cγ([0, T ])

∈ B

(L2([0, T ])

).

Let us prove this. We define for m ∈ N,

Um := f ∈ L2([0, T ]) : ∃ f ∈ f with ‖f‖Cγ ≤ m.

Now fix an m ∈ N and let (fn)∞n=1 ⊆ Um converge to some f ∈ L2([0, T ]). Then

there exists a subsequence (fnk)∞k=1 that converges pointwise almost everywhere

to f . Therefore,∣∣f(t)− f(s)∣∣ = lim

k→∞

∣∣fnk(t)− fnk(s)∣∣ ≤ m|t− s|γ almost everywhere.

The continuous extension f of f then satisfies∣∣f(t)− f(s)∣∣ ≤ m|t− s|γ for all s, t ∈ [0, T ].

26 Random Series

We infer that f ∈ Um, and this implies that Um is closed in L2([0, T ]). Thus, we

have

Cγ([0, T ]) =⋃m∈N

Um ∈ B(L2([0, T ])

).

Let(X(t)

)t∈[0,T ]

be the Brownian motion constructed in Theorem 1.20. Then the

random variables β and X have the same distribution and the Borel measures Pβand PX agree on B

(L2([0, T ])

). Hence, by Theorem 1.20,

P(β ∈ Cγ([0, T ])

)= P

(X ∈ Cγ([0, T ])

)= 1,

which means that(β(t)

)t∈[0,T ]

has a version with γ-Holder continuous trajectories

for any exponent γ < 12.

Chapter 2

Martingale Inequalities

This chapter is devoted to the study of the UMD property and a new version

of Doob’s martingale inequality. Both proves are based on a good-λ inequality

for which it will be necessary to run a reduction process first. The purpose of

this process is to verify that it is sufficient to prove the estimates only for a

special class of martingales. Subsequently, we will apply these results to obtain

a stronger version of the Burkholder-Gundy inequality as well as a Decoupling

theorem. The latter one will play an important role in the construction of a

stochastic integral for processes. But first of all, we take a look at martingale

difference sequences.

In this chapter we may always assume that 1 < r <∞ is fixed.

2.1 Martingale Difference Sequences

We start with the definition.

Definition 2.1. Let (Mn)Nn=1 be an Lr(U)-valued martingale. The sequence

(dn)Nn=1 defined by dn := Mn −Mn−1 (with M0 = 0) is called the martingale

difference sequence associated with (Mn)Nn=1. Additionally, we call (dn)Nn=1

an Lp martingale difference sequence if it is the difference sequence of an Lp

martingale.

Remark 2.2. (1) If (Mn)Nn=1 is a martingale with respect to the filtration (Fn)Nn=1,

then (dn)Nn=1 is adapted to (Fn)Nn=1.

(2) For 1 ≤ m < n ≤ N , we have

E[dn|Fm] = E[Mn|Fm]− E[Mn−1|Fm] = Mm −Mm = 0.

28 Martingale Inequalities

The next proposition shows that the properties of Remark 2.2 already give a

characterization of martingale difference sequences.

Proposition 2.3. Let (dn)Nn=1 be a sequence of integrable Lr(U)-valued ran-

dom variables satisfying the properties of Remark 2.2. Then (Mn)Nn=1 defined

by Mn :=∑n

i=1 di, 1 ≤ n ≤ N , is a martingale with respect to (Fn)Nn=1, and

(dn)Nn=1 is the martingale difference sequence associated with (Mn)Nn=1.

Proof. Let 1 ≤ m < n ≤ N be fixed. Since d1, . . . , dn are Fn-measurable and

integrable, also Mn =∑n

i=1 di is Fn-measurable and integrable. Using (A.2) and

the fact that E[di|Fm] = 0 for all m < i ≤ N , we obtain

E[Mn|Fm] =n∑i=1

E[di|Fm] =m∑i=1

E[di|Fm] =m∑i=1

di = Mm.

Hence, (Mn)Nn=1 is a martingale with respect to the filtration (Fn)Nn=1 and, clearly,

Mn −Mn−1 = dn for each n = 1, . . . , N .

This fact allows us to talk about martingale difference sequences without men-

tioning the associated martingale. Next, we will take a closer look on the case

r = 2, i.e., the Hilbert space case.

Proposition 2.4. Let (dn)Nn=1 be an L2(U)-valued L2 martingale difference

sequence, then the following assertions hold:

(1) E∫Udndm dµ = 0 for all n 6= m;

(2) E∥∥∑N

n=1 εndn∥∥2

2≤ E

∥∥∑Nn=1 dn

∥∥2

2for all sequences (εn)Nn=1 ⊆ [−1, 1].

Proof. (1) By Lemma A.26,(dn(u)

)Nn=1

is an L2 martingale difference sequence

with respect to (Fn)Nn=1 for all u ∈ U outside a µ-null set U0. For each fixed

u ∈ U \ U0 and 1 ≤ m < n ≤ N , we obtain

Edn(u)dm(u) = E(E[dn(u)dm(u)|Fn−1]

)= E

(dm(u)E[dn(u)|Fn−1]

)= 0,

where we used (A.1), (A.4) and the Fn−1-measurability of dm(u), as well as

Remark 2.2 (2). Applying Fubini’s Theorem now yield

E∫U

dndm dµ =

∫U

Edndm dµ = 0.

2.2 The Strong Doob and Strong Burkholder-Gundy Inequality 29

(2) Using (1) and Fubini’s theorem, we obtain

E∥∥∥ N∑n=1

dn

∥∥∥2

2=

N∑n=1

N∑m=1

E∫U

dndm dµ =N∑n=1

E∫U

d2n dµ,

which finally leads to

E∥∥∥ N∑n=1

εndn

∥∥∥2

2=

N∑n=1

ε2nE∫U

d2n dµ ≤

N∑n=1

E∫U

d2n dµ = E

∥∥∥ N∑n=1

dn

∥∥∥2

2.

2.2 The Strong Doob and Strong Burkholder-

Gundy Inequality

Motivated by Proposition 2.4 in the previous section and Doob’s martingale in-

equality (cf. Theorem A.24), we are going to show the following estimates:

Theorem 2.5 (UMD1 property). Let 1 < p < ∞ and (εn)Nn=1 ⊆ [−1, 1].

Then we have for all Lr(U)-valued Lp martingale difference sequences (dn)Nn=1

E∥∥∥ N∑n=1

εndn

∥∥∥pr.p,r E

∥∥∥ N∑n=1

dn

∥∥∥pr. (UMD)

Theorem 2.6 (Strong Doob inequality). Let 1 < p <∞. Then we have

for all Lr(U)-valued Lp martingales (Mn)Nn=1

E∥∥ N

maxn=1|Mn|

∥∥pr.p,r E‖MN‖pr. (SD)

Remark 2.7. Note that

E Nmaxn=1‖Mn‖pr ≤ E

∥∥ Nmaxn=1|Mn|

∥∥pr,

so the result of Theorem 2.6 is stronger than the ’classical’ Doob inequality.

The proof of each estimate consists of two parts: a reduction of the problem

to so called Haar martingales and then proving those estimates for this class of

martingales.

1The term ’UMD’ is an abbreviation for ’unconditional martingale differences’.


2.2.1 The Reduction Process

We consider the reduction process for both estimates simultaneously. There will

be 3 steps:

Step 1: Reduction to divisible probability spaces

A probability space (Ω,A ,P) is said to be divisible if for all A ∈ A and 0 < s < 1

we have A = A1 ∪ A2 with A1, A2 ∈ A and

P(A1) = sP(A), P(A2) = (1− s)P(A).

Lemma 2.8. Let 1 < p <∞. If (UMD) or (SD) hold on any divisible prob-

ability space, then (UMD) or (SD) hold on an arbitrary probability space,

respectively.

Proof. Let (Mn)Nn=1 be an Lr(U)-valued Lp martingale with respect to the fil-

tration (Fn)Nn=1 and let (dn)Nn=1 be its martingale difference sequence, defined on

an arbitrary probability space (Ω,A ,P). We set

Ω := Ω× [0, 1], A := A ⊗B([0, 1]), P := P⊗ λ[0,1].

Then (Ω, A , P) is a divisible probability space. To see this, we define for an

A ∈ A

φA : [0, 1]→ [0, 1], φA(t) = P(A ∩

(Ω× [0, t]

)).

Then, by construction, φA is continuous with φA(1) = P(A) and φA(0) = 0. Now

fix 0 < s < 1. By the intermediate value theorem, there exists a t0 ∈ [0, 1] with

φA(t0) = sP(A). Thus, by taking

A1 := A ∩(Ω× [0, t0]

)and A2 := A \ A1,

we have A1, A2 ∈ A satisfying A1 ∪ A2 = A, P(A1) = sP(A), and P(A2) =

P(A)− P(A1) = (1− s)P(A).

Now define Mn(ω, t) := 1[0,1](t)Mn(ω) for (ω, t) ∈ Ω and Fn := Fn ⊗B([0, 1]).

Clearly, (Mn)Nn=1 is integrable and adapted to (Fn)Nn=1. For 1 ≤ m < n ≤ N ,

using (A.8), we obtain

E[Mn|Fm] = E[1[0,1]Mn|Fm ⊗B([0, 1])

]= E

[1[0,1]|B([0, 1])

]E[Mn|Fm]

= 1[0,1]Mm = Mm.


Therefore, (Mn)Nn=1 is a martingale with respect to (Fn)Nn=1. Now, let (εn)Nn=1 ⊆[−1, 1]. Then, by the assumption, we obtain for dn := Mn − Mn−1

E∥∥∥ N∑n=1

εndn

∥∥∥pr

= E∥∥∥ N∑n=1

εndn

∥∥∥pr.p,r E

∥∥∥ N∑n=1

dn

∥∥∥pr

= E∥∥∥ N∑n=1

dn

∥∥∥pr.

And similarly, we get

E∥∥ N

maxn=1|Mn|

∥∥pr

= E∥∥ N

maxn=1|Mn|

∥∥pr.p,r E‖MN‖pr = E‖MN‖pr.

Step 2: Reduction to dyadic filtrations

Let (Ω,A ,P) be a probability space, and let G be a sub-σ-algebra of A . G is

called dyadic if it is generated by 2m disjoint sets of measure 2−m for an integer

m ≥ 0. Accordingly, we call a filtration in (Ω,A ,P) dyadic if each of its consti-

tuting σ-algebras is dyadic.

We first need a simple approximation result.

Lemma 2.9. Let 1 ≤ p < ∞, ε > 0, and f be an Lr(U)-valued simple

random variable on a divisible probability space (Ω,A ,P). Let G be a dyadic

sub-σ-algebra of A . Then there exists a dyadic sub-σ-algebra G ⊆H ⊆ A

and an H -measurable simple random variable h satisfying(E‖h−f‖pr

) 1p < ε.

Proof. (1) Suppose that G is generated by the 2m sets (Gi)2m

i=1 with P(Gi) = 2−m

for all i ∈ 1, . . . , 2m. We first prove the lemma for an indicator function f = 1A

with A ∈ A fixed. Write 1A =∑2m

i=1 1A∩Gi , and, for an arbitrary generating set

Gk, let (bGkj )∞j=1 be the sequence of digits in the binary expansion of the real

number P(A ∩Gk), i.e.

P(A ∩Gk) =∞∑j=1

bGkj 2−j.

Then, set AGk0 := A ∩ Gk and BGk0 := ∅, and for j ≥ 1, if bGkj = 1, choose

BGkj ⊆ AGkj−1 satisfying BGk

j ∈ A and P(BGkj ) = bGkj 2−j. If bGkj = 0, take BGk

j = ∅.Then set AGkj := AGkj−1\B

Gkj and continue. The so constructed sets (BGk

j )∞j=1 ⊆ A

are disjoint, contained in Gk, and satisfy

P(

(A ∩Gk) \∞⋃j=1

BGkj

)= P(A ∩Gk)− P

( ∞⋃j=1

BGkj

)= P(A ∩Gk)−

∞∑j=1

P(BGkj )

= P(A ∩Gk)−∞∑j=1

bGkj 2−j = 0.


Let nk ≥ 1 be the first integer with

P(

(A ∩Gk) \nk⋃j=1

BGkj

)≤( ε

2m

)p.

For every 1 ≤ j ≤ nk with bGkj = 1 we have P(BGkj ) = 2−j. It follows that we

can split Gk into disjoint subsets of measure 2−nk such that every BGkj is a finite

union of these subsets.

We now repeat this construction for each of the 2m generating sets (Gi)2m

i=1. Then

every Gi is divided into sets of measure 2−ni . Set N := max2m

i=1 ni, and, if nec-

essary, subdivide the generating sets even further such that every Gi is divided

into sets of measure 2−N . Let H be the σ-algebra generated by the 2N sets of

measure 2−N thus obtained. This σ-algebra is dyadic with G ⊆ H , and the

function

h :=2m∑i=1

N∑j=1,b

Gij =1

1BGij

is H -measurable with

(E|f − h|p

) 1p =

(E∣∣∣ 2m∑i=1

1A∩Gi −2m∑i=1

N∑j=1,b

Gij =1

1BGij

∣∣∣p) 1p

=

(E∣∣∣ 2m∑i=1

1(A∩Gi)\

⋃Nj=1B

Gij

∣∣∣p) 1p

≤2m∑i=1

P(

(A ∩Gi) \N⋃j=1

BGij

) 1p

≤2m∑i=1

ε

2m= ε.

(2) For the general case, we may assume that at least one xn is not 0 and define

ε := ε(∑N

n=1 ‖xn‖r)−1

. By (1) we can find for every indicator function 1An

an H -measurable indicator function 1Bn with(E|1An − 1Bn|p

) 1p < ε. Taking

h :=∑N

n=1 1Bnxn finally yield

(E‖h− f‖pr

) 1p ≤

N∑n=1

(E|1An − 1Bn|p‖xn‖pr

) 1p <

N∑n=1

‖xn‖rε = ε.

With this preparation we can finish step 2.


Lemma 2.10. Let 1 < p < ∞. If (UMD) or (SD) hold on a divisible

probability space (Ω,A ,P) for all Lr(U)-valued Lp martingales with respect

to a dyadic filtration, then (UMD) or (SD) hold for all Lr(U)-valued Lp

martingales on an arbitrary probability space, respectively.

Proof. Let ε > 0 and (Mn)Nn=1 be an arbitrary Lr(U)-valued Lp martingale with

respect to the filtration (Fn)Nn=1 on a divisible probability space (Ω,A ,P). Let

(dn)Nn=1 be its martingale difference sequence. Since every Mn is Fn-measurable

and separably valued, we can find an Fn-measurable simple function sn : Ω →Lr(U) with

(E ‖Mn − sn‖pr

) 1p ≤ ε

4N, using Proposition A.3. By repeated applica-

tions of Lemma 2.9 we can find a sequence of dyadic sub-σ-algebras (Fn)Nn=1 with

F0 = ∅,Ω and Fn−1 ⊆ Fn ⊆ Fn, and a sequence of Fn-measurable simple

functions (sn)Nn=1 with(E ‖sn − sn‖pr

) 1p ≤ ε

4N.

For n ≥ 1, define Mn := E[Mn|Fn]. Then Mn is Fn-measurable, integrable, and

E[Mn|Fn−1] = E[E[Mn|Fn]

∣∣Fn−1

]= E[Mn|Fn−1]

= E[E[Mn|Fn−1]

∣∣Fn−1

]= E[Mn−1|Fn−1]

= Mn−1.

Therefore, (Mn)Nn=1 is a martingale with respect to (Fn)Nn=1. Finally, by the

Lp-contractivity of the conditional expectation operator (cf. Theorem A.21), we

obtain(E‖Mn − Mn‖pr

) 1p ≤

(E‖Mn − sn‖pr

) 1p +

(E‖sn − sn‖pr

) 1p +

(E‖sn − Mn‖pr

) 1p

≤ ε

2N+(E∥∥E[sn|Fn]− E[Mn|Fn]

∥∥pr

) 1p

=ε

2N+(E∥∥E[sn −Mn|Fn]

∥∥pr

) 1p

≤ ε

2N+(E‖sn −Mn‖pr

) 1p

≤ ε

2N+

ε

2N=

ε

N.

Moreover, if (dn)Nn=1 is the martingale difference sequence associated with (Mn)Nn=1,

then, (E‖dn − dn‖pr

) 1p ≤

(E‖Mn − Mn‖pr

) 1p +

(E‖Mn−1 − Mn−1‖pr

) 1p

≤ 2ε

N.


Finally, by the assumptions and the Lp-contractivity, we get(E∥∥∥ N∑n=1

εndn

∥∥∥pr

) 1p

.p,r

(E∥∥∥ N∑n=1

dn

∥∥∥pr

) 1p

=(E‖MN‖pr

) 1p =

(E∥∥E[MN |FN ]

∥∥pr

) 1p

≤(E‖MN‖pr

) 1p =

(E∥∥∥ N∑n=1

dn

∥∥∥pr

) 1p

,

and similarly(E∥∥ N

maxn=1|Mn|

∥∥pr

) 1p.p,r

(E‖MN‖pr

) 1p =

(E∥∥E[MN |FN ]

∥∥pr

) 1p

≤(E‖MN‖pr

) 1p .

This leads to(E∥∥∥ N∑n=1

εndn

∥∥∥pr

) 1p

≤(E∥∥∥ N∑n=1

εn(dn − dn)∥∥∥pr

) 1p

+

(E∥∥∥ N∑n=1

εndn

∥∥∥pr

) 1p

.p,r

N∑n=1

|εn|(E‖dn − dn‖pr

) 1p +

(E∥∥∥ N∑n=1

dn

∥∥∥pr

) 1p

≤ 2ε+

(E∥∥∥ N∑n=1

dn

∥∥∥pr

) 1p

,

and (E∥∥ N

maxn=1|Mn|

∥∥pr

) 1p ≤

(E∥∥ N

maxn=1|Mn − Mn|

∥∥pr

) 1p

+(E∥∥ N

maxn=1|Mn|

∥∥pr

) 1p

.p,r

N∑n=1

(E‖Mn − Mn‖pr

) 1p +

(E‖MN‖pr

) 1p

≤ ε+(E‖MN‖pr

) 1p .

Since ε > 0 was arbitrary, this shows the result together with Lemma 2.8.

Step 3: Reduction to Haar filtrations

An atom of a σ-algebra G is a nonempty set G ∈ G such that H ⊆ G with

H ∈ G implies H ∈ ∅, G. In the final step we now want to shrink the class of

martingales to Haar martingales, which are defined as martingales with respect

to a Haar filtration. This is a filtration (Fn)Nn=1 where F1 = ∅,Ω and for

n ≥ 1 each Fn is obtained from Fn−1 by dividing precisely one atom of Fn−1

of maximal measure into two sets of equal measure. By construction, each Fn

is generated by n atoms of measure 2−k−1 or 2−k, where k is the unique integer

such that 2k−1 < n ≤ 2k.


Lemma 2.11. Let 1 < p < ∞. If (UMD) or (SD) hold on a divisible

probability space and for all Lr(U)-valued Lp martingales with respect to a

Haar filtration, then (UMD) or (SD) hold for all Lr(U)-valued Lp martingales

on an arbitrary probability space, respectively.

Proof. Let (Mn)Nn=1 be an Lr(U)-valued Lp martingale with respect to a dyadic

filtration (Fn)Nn=1 on a divisible probability space (Ω,A ,P), and let (dn)Nn=1 be

its martingale difference sequence. We now want to construct a Haar martingale

(Mk)Kk=1 in which we can ’embed’ (Mn)Nn=1.

Each Fn is dyadic, and therefore it is generated by kn := 2ln atoms of measure

2−ln . Since each atom of Fn−1 is a finite union of atoms in Fn, we have k1 <

. . . < kN . Now set k0 := 1, Fk0 := ∅,Ω, and Fkn := Fn. The σ-algebras Fk

with kn−1 < k < kn can now be constructed by splitting the atoms of Fkn−1 one

by one into two disjoint subsets of equal measure so as to arrive at the atoms of

Fkn by repeating this procedure kn − kn−1 times. Completing this process, we

get a Haar filtration (Fk)Kk=1 with K = kN = 2lN . Now take

Mkn := Mn and Mk := E[Mkn|Fk] = E[Mn|Fk] if kn−1 < k < kn.

Then Mk is Fk-measurable, integrable, and satisfies for 1 ≤ m < k ≤ K

E[Mk|Fm] = E[E[Mkn|Fk]

∣∣Fm

]= E[Mkn|Fm] = Mm.

Therefore, (Mk)Kk=1 is a Haar martingale. Let (dk)

Kk=1 be the martingale difference

sequence associated with (Mk)Kk=1. Then,

kn∑j=kn−1+1

dj = Mkn − Mkn−1 = Mn −Mn−1 = dn.

Now let (εn)Nn=1 ⊆ [−1, 1] and set εk := εn for k = kn−1 + 1, . . . , kn, n = 1, . . . , N .

This yield

E∥∥∥ N∑n=1

εndn

∥∥∥pr

= E∥∥∥ N∑n=1

kn∑j=kn−1+1

εndj

∥∥∥pr

= E∥∥∥ K∑k=1

εkdk

∥∥∥pr

.p,r E∥∥∥ K∑k=1

dk

∥∥∥pr

= E∥∥∥ N∑n=1

kn∑j=kn−1+1

dj

∥∥∥pr

= E∥∥∥ N∑n=1

dn

∥∥∥pr.


Finally, observe that MK = MkN = MN , which gives us

E∥∥ N

maxn=1|Mn|

∥∥pr≤ E

∥∥ Kmaxk=1|Mk|

∥∥pr.p,r E‖MK‖pr = E ‖MN‖pr .

Together with Lemma 2.10, this completes the prove.

At the end of this subsection we want to show a special property of Haar mar-

tingales.

Lemma 2.12. Let (Mn)Nn=1 be an Lr(U)-valued Haar martingale and let

(dn)Nn=1 be its martingale difference sequence. Then ‖dn+1‖r is Fn-measurable

for all n = 1, . . . , N − 1.

Proof. Note that for any m = 1, . . . , N , we have

Fm = σ(A1, . . . , Am) =⋃j∈M

Aj : M ⊆ 1, . . . ,m.

Fix k ∈ 1, . . . ,m, choose an arbitrary ω ∈ Ak, and set Mm(ω) = z. By the

Fm-measurability of Mm we get M−1m (z) ∈ Fm. Therefore, there exists an

A ∈ Fm with Mm(A) = z, and, by the structure of Fm, it holds Ak ⊆ A.

From this we infer that Mm is constant on every generating atom of Fm. Now

fix an n ∈ 1, . . . , N − 1. Fn+1 is obtained by splitting one generating atom

B ∈ Fn into two subsets B1 and B2 of equal measure. By the foregoing remark,

Mn and Mn+1 only differ on B. In fact, let C be one of the generating atoms

of Fn, unequal to B. Then C is also a generating atom of Fn+1, and Mn = z1,

Mn+1 = z2 on C for some z1, z2 ∈ Lr(U). Hence, by Theorem A.21,

P(C)(z2 − z1) =

∫C

dn+1 dP =

∫C

E[dn+1|Fn] dP = 0.

Since C has positive measure, this implies z1 = z2, and hence, Mn+1 and Mn are

equal on each generating atom of Fn, unequal to B.

Also, dn+1 is constant on B1 and B2 with values x1 and x2. Then,

P(B1)x1 + P(B2)x2 =

∫B

dn+1 dP =

∫B

E[dn+1|Fn] dP = 0,

and from P(B1) = P(B2) > 0 we deduce that x1 = −x2. Therefore,

‖dn+1‖r = 1B1‖x1‖r + 1B2‖x2‖r = 1B‖x1‖r

is Fn-measurable.


2.2.2 The UMD Property

By subsection 2.2.1 it suffices to consider Haar martingales in order to prove

Theorem 2.5. Thus, for the rest of this subsection, let (Mn)Nn=1 be an Lr(U)-

valued Lp martingale with respect to the Haar filtration (Fn)Nn=1, and let (dn)Nn=1

be its martingale difference sequence. For a fixed sequence (εn)Nn=1 ⊆ [−1, 1] we

denote by (gn)Nn=1 the martingale transform gn :=∑n

j=1 εjdj, and we define

M∗(ω) :=N

maxn=1‖Mn(ω)‖r, g∗(ω) :=

Nmaxn=1‖gn(ω)‖r.

Remark 2.13. If (Xn)Nn=1 is a sequence of Lr(U)-valued random variables and

τ : Ω→ 1, . . . , N is another random variable, then we define

Xτ : Ω→ Lr(U), Xτ (ω) := Xτ(ω)(ω).

Note that Xτ is still measurable since Xτ =∑N

n=1 1τ=nXn.

Lemma 2.14. Suppose that (UMD) holds for some 1 < q < ∞. Then we

have for all δ > 0, β > 2δ + 1, and all λ > 0

P(g∗ > βλ, M∗ ≤ δλ) ≤ α(δ)q P(g∗ > λ),

where α(δ) := 4δc(q,r)β−2δ−1

and c(q, r) is the constant from Theorem 2.5.

Proof. We define

µ(ω) := min1 ≤ n ≤ N : ‖gn(ω)‖r > λ,ν(ω) := min1 ≤ n ≤ N : ‖gn(ω)‖r > βλ and

σ(ω) := min1 ≤ n ≤ N : ‖Mn(ω)‖r > δλ or ‖dn+1(ω)‖r > 2δλ

with the convention that min ∅ := N + 1 and dN+1 := 0.

It holds that µ = j, ν = j ∈ Fj and, by Lemma 2.12, even σ = j ∈ Fj

for all j = 1, . . . , N . Now define for n = 1, . . . , N

vn := 1µ<n≤ν∧σ,

and note that

µ < n ≤ ν ∧ σ = µ < n ∩ ν ∧ σ ≥ n= µ < n ∩ ν ∧ σ < nC ∈ Fn−1.


Therefore, (vn)Nn=1 is predictable with respect to (Fn)Nn=1, and so, by Example

A.23,

Vn :=n∑j=1

vjdj

defines a martingale (Vn)Nn=1, which is adapted to (Fn)Nn=1.

On the set σ ≤ µ we have vn = 0 for all n = 1, . . . , N , which means that VN = 0

there. Now let ω ∈ σ > µ. Then ‖Mµ(ω)‖r ≤ δλ. Also, if ν(ω) ∧ σ(ω) > 1,

then from ‖M(ν∧σ)−1(ω)‖r ≤ δλ and ‖dν∧σ(ω)‖r ≤ 2δλ it follows that

‖Mν∧σ(ω)‖r ≤ ‖M(ν∧σ)−1(ω)‖r + ‖dν∧σ(ω)‖r ≤ 3δλ.

If ν(ω) ∧ σ(ω) = 1, then, since µ(ω) ≥ 1, we again have VN(ω) = 0. Hence, on

the set σ > µ we obtain

‖VN‖r =∥∥∥ ∑µ<j≤ν∧σ

dj

∥∥∥r

= ‖Mν∧σ −Mµ‖r ≤ 4δλ.

Note that g∗ ≤ λ = µ = N + 1 ⊆ σ ≤ µ, and thus σ > µ ⊆ g∗ > λ.We infer that

E‖VN‖qr ≤ (4δλ)q P(σ > µ) ≤ (4δλ)q P(g∗ > λ).

Next, consider the martingale transform (V εn )Nn=1 defined by

V εn :=

n∑j=1

εjvjdj.

If ω ∈ ν ≤ N, σ = N+1, we obtain ν(ω)∧σ(ω) = ν(ω) as well as ‖gν(ω)‖r > βλ

and ‖dn(ω)‖r ≤ 2δλ for all n = 2, . . . , N + 1 and ‖d1(ω)‖r = ‖M1(ω)‖r ≤ δλ.

Therefore, if µ(ω) > 1,

‖gµ(ω)‖r ≤ ‖gµ−1(ω)‖r + ‖dµ(ω)‖r ≤ λ+ 2δλ.

and if µ(ω) = 1,

‖gµ(ω)‖r = ‖g1(ω)‖r ≤ ‖M1(ω)‖r ≤ δλ.

Hence, on the set ν ≤ N, σ = N + 1 we get in any case

‖V εN‖r =

∥∥∥ ∑µ<j≤ν

εjdj

∥∥∥r

= ‖gν − gµ‖r > βλ− 2δλ− λ.


Observe that g∗ > βλ = ν ≤ N and M∗ ≤ δλ = σ = N + 1. Then

Chebyshev’s inequality yield

P(g∗ > βλ, M∗ ≤ δλ) = P(ν ≤ N, σ = N + 1)

≤ P(‖V ε

N‖r > βλ− 2δλ− λ)

≤ 1

(βλ− 2δλ− λ)qE‖V ε

N‖qr

≤ c(q, r)q

(βλ− 2δλ− λ)qE‖VN‖qr

≤ (4δ)qc(q, r)q

(β − 2δ − 1)qP(g∗ > λ).

Theorem 2.15. If (UMD) holds for some 1 < q < ∞, then it holds for all

1 < p <∞.

Proof. By the previous subsection and the definitions made before Lemma 2.14,

we need to prove the estimate

E‖gN‖pr ≤ bp E‖MN‖pr

with a constant b ≥ 0 depending only on p, q and r.

Note that for any A, B ∈ A we have

P(A) = P(A ∩B) + P(A ∩BC) ≤ P(A ∩B) + P(BC).

Now fix a β > 1, and let δ > 0 be so small such that β > 2δ + 1. Then, by

Lemma 2.14,

P(g∗ > βλ) ≤ P(g∗ > βλ, M∗ ≤ δλ) + P(M∗ > δλ)

≤ α(δ)qP(g∗ > λ) + P(M∗ > δλ).

Therefore, by Proposition 1.2 and Doob’s inequality, we obtain

E‖gN‖pr ≤ E|g∗|p =

∫ ∞0

pλp−1P(g∗ > λ) dλ = βp∫ ∞

0

pλp−1P(g∗ > βλ) dλ

≤ α(δ)qβp∫ ∞

0

pλp−1P(g∗ > λ) dλ+ βp∫ ∞

0

pλp−1P(M∗ > δλ) dλ

= α(δ)qβpE|g∗|p +βp

δpE|M∗|p

≤ α(δ)qβp( p

p− 1

)pE‖gN‖pr +

βp

δp

( p

p− 1

)pE‖MN‖pr.


Since limδ→0 α(δ) = 0, we may arrange that α(δ)qβp(

pp−1

)p< 1 by taking δ > 0

small enough. Recalling that (Mn)Nn=1 is an Lp martingale, we note that E‖gN‖pr <∞. Hence,

E‖gN‖pr ≤βp(

pp−1

)p(1−

(pp−1

)pα(δ)qβp

)δpE‖MN‖pr.

Remark 2.16. By combining Proposition 2.4 and Theorem 2.15, we obtain for

all 1 < p < ∞, all R-valued Lp martingale difference sequences (dn)Nn=1, and all

sequences (εn)Nn=1 ⊆ [−1, 1]

E∣∣∣ N∑n=1

εndn

∣∣∣p .p E∣∣∣ N∑n=1

dn

∣∣∣p.

Now we can prove the main result of this subsection.

Proof (of Theorem 2.5). Let (dn)Nn=1 be an arbitrary Lr(U)-valued Lp martin-

gale difference sequence and (εn)Nn=1 ⊆ [−1, 1]. Then, by Lemma A.26, (dn(u))Nn=1

is an R-valued martingale difference sequence for µ-almost all u ∈ U . Thus, by

the foregoing remark,

E∣∣∣ N∑n=1

εndn(u)∣∣∣r .r E∣∣∣ N∑

n=1

dn(u)∣∣∣r

for µ-almost all u ∈ U . Applying Fubini’s theorem now yield

E∥∥∥ N∑n=1

εndn

∥∥∥rr

=

∫U

E∣∣∣ N∑n=1

εndn

∣∣∣r dµ .r

∫U

E∣∣∣ N∑n=1

dn

∣∣∣r dµ = E∥∥∥ N∑n=1

dn

∥∥∥rr.

Hence, (UMD) holds for 1 < r < ∞, and thanks to Theorem 2.15, the UMD

property holds for all 1 < p <∞.

Remark 2.17. If we restrict the choice of the sequence (εn)Nn=1 to +1,−1,then we get even more. If (dn)Nn=1 is an Lr(U)-valued Lp martingale difference

sequence, then the same is true for (εndn)Nn=1. Hence, by applying Theorem 2.5,

we obtain

E∥∥∥ N∑n=1

dn

∥∥∥pr

= E∥∥∥ N∑n=1

εn(εndn)∥∥∥pr.p,r E

∥∥∥ N∑n=1

εndn

∥∥∥pr.

So, in this case we get

E∥∥∥ N∑n=1

εndn

∥∥∥prhp,r E

∥∥∥ N∑n=1

dn

∥∥∥pr.


2.2.3 The Strong Doob Inequality

By the reduction process of subsection 2.2.1 it is enough to consider Haar martin-

gales to prove Theorem 2.6. In what follows, we let (Mn)Nn=1 be an Lr(U)-valued

Lp martingale with respect to a Haar filtration (Fn)Nn=1 and with difference se-

quence (dn)Nn=1. Then we define

M∗(ω) :=N

maxn=1‖Mn(ω)‖r, M∗(ω) :=

∥∥ Nmaxn=1|Mn(ω)|

∥∥r.

Lemma 2.18. Let (Vn)Nn=1 be an Lr(U)-valued Lr martingale. Then we have

for all λ > 0

P(∥∥ N

maxn=1|Vn|

∥∥r> λ

)≤

( rr−1

)r

λrE‖VN‖rr.

Proof. By Lemma A.26,(Vn(u)

)Nn=1

is an R-valued Lr martingale for µ-almost

all u ∈ U . Thus, by Doob’s martingale inequality, we obtain

E Nmaxn=1|Vn(u)|r ≤

( r

r − 1

)rE|VN(u)|r

for µ-almost all u ∈ U . Define

V ∗ :=∥∥ N

maxn=1|Vn|

∥∥r.

Then Fubini’s theorem yield

λrP(V ∗ > λ

)≤ E

(1V ∗>λV

∗ r) ≤ EV ∗ r

= E(∫

U

Nmaxn=1|Vn|r dµ

)=

∫U

(E N

maxn=1|Vn|r

)dµ

≤( r

r − 1

)r ∫U

E|VN |r dµ =( r

r − 1

)rE∫U

|VN |r dµ

=( r

r − 1

)rE‖VN‖rr.

Lemma 2.19. For all δ > 0, β > 2δ + 1, and all λ > 0 we have

P(M∗ > βλ, M∗ ≤ δλ

)≤ α(δ)rP

(M∗ > λ

),

where α(δ) := rr−1

4δβ−2δ−1

.


Proof. We define

µ(ω) := min

1 ≤ n ≤ N :∥∥ n

maxj=1|Mj(ω)|

∥∥r> λ

,

ν(ω) := min

1 ≤ n ≤ N :∥∥ n

maxj=1|Mj(ω)|

∥∥r> βλ

and

σ(ω) := min

1 ≤ n ≤ N : ‖Mn(ω)‖r > δλ or ‖dn+1(ω)‖r > 2δλ

with the understanding that min ∅ := N + 1 and dN+1 := 0. Since β > 1, we

have µ ≤ ν. Observe that µ = j, ν = j ∈ Fj and, by Lemma 2.12, also

σ = j ∈ Fj for all j = 1, . . . , N . For n = 1, . . . , N we set

vn := 1µ<n≤ν∧σ.

Since

µ < n ≤ ν ∧ σ = µ < n ∩ ν ∧ σ ≥ n= µ < n ∩ ν ∧ σ < nC ∈ Fn−1,

(vn)Nn=1 is an (Fn)Nn=1-predictable sequence of bounded random variables. There-

fore, by Example A.23,

Vn :=n∑j=1

vjdj

defines a martingale (Vn)Nn=1, which is adapted to (Fn)Nn=1.

On the set σ ≤ µ we have vn = 0 for all n = 1, . . . , N , which means that VN = 0

there. Now let ω ∈ σ > µ. Then ‖Mµ(ω)‖r ≤ δλ. Also, if ν(ω) ∧ σ(ω) > 1,

then from

‖M(ν∧σ)−1(ω)‖r ≤ δλ and ‖dν∧σ(ω)‖r ≤ 2δλ

it follows that

‖Mν∧σ(ω)‖r ≤ ‖M(ν∧σ)−1(ω)‖r + ‖dν∧σ(ω)‖r ≤ 3δλ.

If ν(ω) ∧ σ(ω) = 1, then, since µ(ω) ≥ 1, we again have VN(ω) = 0. Hence, on

the set σ > µ we obtain

‖VN‖r =∥∥∥ ∑µ<j≤ν∧σ

dj

∥∥∥r

= ‖Mν∧σ −Mµ‖r ≤ 4δλ.

Note that M∗ ≤ λ = µ = N + 1 ⊆ σ ≤ µ, and thus σ > µ ⊆ M∗ > λ.We deduce that

E‖VN‖rr ≤ (4δλ)r P(σ > µ) ≤ (4δλ)r P(M∗ > λ

).


On the set ν ≤ N, σ = N + 1 we have ν ∧ σ = ν, and therefore

Vn =∑

µ<j≤ν∧n

Mj −Mj−1 =

0, if 1 ≤ n ≤ µ,

Mn −Mµ, if µ < n ≤ ν,

Mν −Mµ, if n > ν.

We infer that ∥∥ Nmaxn=1|Vn|

∥∥r

=∥∥ maxµ<n≤ν

|Mn −Mµ|∥∥r.

Now fix a u ∈ U . Then

max1≤n≤ν

|Mn(u)| ≤ max1≤n≤µ

|Mn(u)|+ maxµ<n≤ν

|Mn(u)|,

and

max1≤n≤µ

|Mn(u)| ≤ max1≤n<µ

|Mn(u)|+ |Mµ(u)|.

Using these inequalities, we obtain

maxµ<n≤ν

|Mn(u)−Mµ(u)| ≥ maxµ<n≤ν

|Mn(u)| − |Mµ(u)|

≥ max1≤n≤ν

|Mn(u)| − max1≤n≤µ

|Mn(u)| − |Mµ(u)|

≥ max1≤n≤ν

|Mn(u)| − max1≤n<µ

|Mn(u)| − 2|Mµ(u)|.

It follows that∥∥ Nmaxn=1|Vn|

∥∥r≥∥∥max

1≤n≤ν|Mn|

∥∥r−∥∥ max

1≤n<µ|Mn|

∥∥r− 2‖Mµ‖r.

Next, observe that σ = N + 1 implies that ‖Mn‖r ≤ δλ for all n = 1, . . . , N , and

from ν ≤ N it follows that∥∥max1≤n≤ν |Mn|

∥∥r> βλ and

∥∥max1≤n<µ |Mn|∥∥r≤ λ.

Therefore, we have ∥∥ Nmaxn=1|Vn|

∥∥r> βλ− λ− 2δλ.

Observe that M∗ > βλ = ν ≤ N and M∗ ≤ δλ = σ = N + 1. Putting

all these estimates together and using Lemma 2.18, we get

P(M∗ > βλ, M∗ ≤ δλ

)= P(ν ≤ N, σ = N + 1)

≤ P(∥∥ N

maxn=1|Vn|

∥∥r> βλ− 2δλ− λ

)≤

( rr−1

)r

(βλ− 2δλ− λ)rE‖VN‖rr

≤(4δ)r( r

r−1)r

(β − 2δ − 1)rP(M∗ > λ

).


Proof (of Theorem 2.6). By subsection 2.2.1 and the definitions we made be-

fore Lemma 2.18, we have to show that

E|M∗|p ≤ cp E‖MN‖pr

with a constant c ≥ 0 depending only on p and r.

Note that for any A, B ∈ A we have

P(A) = P(A ∩B) + P(A ∩BC) ≤ P(A ∩B) + P(BC).

Now fix a β > 1, and let δ > 0 be so small such that β > 2δ + 1. Then, by

Lemma 2.19, we have

P(M∗ > βλ

)≤ P

(M∗ > βλ, M∗ ≤ δλ

)+ P(M∗ > δλ)

≤ α(δ)rP(M∗ > λ

)+ P(M∗ > δλ).

Thus, by Proposition 1.2 and Doob’s martingale inequality, we obtain

E|M∗|p =

∫ ∞0

pλp−1P(M∗ > λ

)dλ

= βp∫ ∞

0

pλp−1P(M∗ > βλ

)dλ

≤ α(δ)rβp∫ ∞

0

pλp−1P(M∗ > λ

)dλ+ βp

∫ ∞0

pλp−1P(M∗ > δλ) dλ

= α(δ)rβpE|M∗|p +βp

δpE |M∗|p

≤ α(δ)rβpE|M∗|p +βp

δp

( p

p− 1

)pE ‖MN‖pr .

Since limδ→0 α(δ) = 0, we may take δ > 0 so small such that α(δ)rβp < 1. By

recalling that (Mn)Nn=1 is an Lp martingale, we note that E|M∗|p <∞. Hence we

get

E|M∗|p ≤βp(

pp−1

)p(1− α(δ)rβp) δp

E ‖MN‖pr .

Remark 2.20. Note that ‖MN‖r ≤∥∥maxNn=1 |Mn|

∥∥r

almost surely. Therefore,

we have

E‖MN‖pr ≤ E∥∥ N

maxn=1|Mn|

∥∥pr,

and together with Theorem 2.6 this gives

E∥∥ N

maxn=1|Mn|

∥∥prhp,r E‖MN‖pr.


2.2.4 The Strong Burkholder-Gundy Inequality

Theorem 2.21. Let 1 < p <∞ and (dn)Nn=1 be an Lr(U)-valued Lp martin-

gale difference sequence. Then we have

E∥∥∥ N∑n=1

dn

∥∥∥prhp,r E

∥∥∥( N∑n=1

|dn|2) 1

2∥∥∥pr.

Proof. Let (Ω′,A ′,P′) be another probability space and (r′n)Nn=1 be a sequence

of independent Rademacher variables on (Ω′,A ′,P′). Then, by the Kahane in-

equality, we get for each fixed ω ∈ Ω

∥∥∥( N∑n=1

|dn(ω)|2) 1

2∥∥∥prhp,r E′

∥∥∥ N∑n=1

r′ndn(ω)∥∥∥pr.

Note that(r′n(ω′)

)Nn=1⊆ +1,−1 for all ω′ ∈ Ω′. Therefore, by Fubini’s theorem

and Remark 2.17,

E∥∥∥( N∑

n=1

|dn|2) 1

2∥∥∥prhp,r EE′

∥∥∥ N∑n=1

r′ndn

∥∥∥pr

= E′E∥∥∥ N∑n=1

r′ndn

∥∥∥pr

hp,r E′E∥∥∥ N∑n=1

dn

∥∥∥pr

= E∥∥∥ N∑n=1

dn

∥∥∥pr.

Theorem 2.22 (Strong Burkholder-Gundy inequality). Let 1 < p <

∞ and (Mn)Nn=1 be an Lr(U)-valued Lp martingale with difference sequence

(dn)Nn=1. Then we have

E∥∥ N

maxn=1|Mn|

∥∥prhp,r E

∥∥∥( N∑n=1

|dn|2) 1

2∥∥∥pr.

Proof. By combining Remark 2.20 and Theorem 2.21, we get

E∥∥ N

maxn=1|Mn|

∥∥prhp,r E‖MN‖pr = E

∥∥∥ N∑n=1

dn

∥∥∥prhp,r E

∥∥∥( N∑n=1

|dn|2) 1

2∥∥∥pr.


2.3 Decoupling

This section is an application of Theorem 2.5 we presented in Subsection 2.2.2

and will play a central role in Chapter 3.

Let 1 < p, r < ∞ be fixed, (Fn)n=1 be a filtration on (Ω,A ,P), and (ξn)Nn=1 be

a sequence of centered integrable random variables in Lp(Ω). Assume that

(1) ξn is Fn-measurable for all 1 ≤ n ≤ N , and

(2) ξn is independent of Fm for all 1 ≤ m < n ≤ N .

Note that E [ξn|Fm] = Eξn = 0. Hence, by Proposition 2.3, (ξn)Nn=1 is a martin-

gale difference sequence with respect to (Fn)Nn=1.

On the product space (Ω× Ω,A ⊗A ,P⊗ P) we define

ξn(ω, ω) := ξn(ω), ξn(ω, ω) := ξn(ω).

The sequences (ξn)Nn=1 and (ξn)Nn=1 are independent, identically distributed, and

martingale difference sequences on Ω×Ω with respect to the filtrations (Fn)Nn=1

and (Fn)Nn=1 defined by

Fn := Fn ⊗ ∅,Ω , Fn := ∅,Ω ⊗Fn.

Let (vn)Nn=1 be an (Fn)Nn=1-predictable sequence of Lr(U)-valued random vari-

ables. In the same way as above, we identify (vn)Nn=1 with a predictable sequence

(vn)Nn=1 on Ω× Ω by taking vn(ω, ω) := vn(ω).

Theorem 2.23 (Decoupling). Under the assumptions taken above, we have

EE∥∥∥ N∑n=1

ξnvn

∥∥∥prhp,r EE

∥∥∥ N∑n=1

ξnvn

∥∥∥pr.

Proof. For n = 1, . . . , N , define

d2n−1 := 12(ξn + ξn)vn and d2n := 1

2(ξn − ξn)vn,

as well as

G2n−1 := σ(Fn−1, Fn−1, ξn + ξn) and G2n := σ(Fn, Fn).

Then, (Gj)2Nj=1 is a filtration and dj is Gj-measurable for all j = 1, . . . , 2N . Note

that the independence of ξn and Fn−1 implies the independence of σ(ξn+ ξn) and

2.3 Decoupling 47

σ(Fn−1, Fn−1). Now, using the Fn−1-measurability of vn, (A.7), and (A.9), we

obtain

E[d2n|G2n−1] = 12vnE[ξn − ξn|G2n−1] = 1

2vnE[ξn − ξn|ξn + ξn] = 0.

Moreover,

E[d2n−1|G2n−2] = 12vnE[ξn + ξn|Fn−1, Fn−1] = 1

2vnE(ξn + ξn) = 0,

since ξn + ξn is independent of σ(Fn−1, Fn−1) and ξn, ξn are centered. Therefore,

by Proposition 2.3, (dj)2Nj=1 is a martingale difference sequence with respect to the

filtration (Gj)2Nj=1.

Now observe that2N∑j=1

dj =N∑n=1

d2n−1 + d2n =N∑n=1

ξnvn,

and2N∑j=1

(−1)j+1dj =N∑n=1

d2n−1 − d2n =N∑n=1

ξnvn.

Finally, by Remark 2.17, we obtain

EE∥∥∥ N∑n=1

ξnvn

∥∥∥pr

= EE∥∥∥ 2N∑j=1

dj

∥∥∥prhp,r EE

∥∥∥ 2N∑j=1

(−1)j+1dj

∥∥∥pr

= EE∥∥∥ N∑n=1

ξnvn

∥∥∥pr.

As a combination of the Decoupling theorem and the Kahane inequality, we get

the following corollary.

Corollary 2.24. Let(β(t)

)t∈[0,T ]

be an arbitrary Brownian motion, and let

0 = t0 < . . . < tN = T be fixed. Let (Fn)Nn=1 be a filtration which satisfies

(1) β(tn) is Fn-measurable,

(2) β(tn)− β(tn−1) is independent of Fn−1,

and let (vn)Nn=1 be an (Fn)Nn=1-predictable sequence of Lr(U)-valued random

variables. Then we have

E∥∥∥ N∑n=1

vn(β(tn)− β(tn−1)

) ∥∥∥prhp,r E

∥∥∥( N∑n=1

|vn|2 (tn − tn−1)) 1

2∥∥∥pr.


Proof. For n = 1, . . . , N we define

ξn := β(tn)− β(tn−1) and γn :=1

(tn − tn−1)12

(β(tn)− β(tn−1)

).

Then (ξn)Nn=1 satisfies the assumptions of the Decoupling theorem, and, by the

definition of the Brownian motion, (γn)Nn=1 is a sequence of independent standard

Gaussian variables. Using the Kahane inequality, the Decoupling theorem and

the notions given there, we obtain

E∥∥∥ N∑n=1


) ∥∥∥pr

= EE∥∥∥ N∑n=1

vnξn

∥∥∥pr

hp,r EE∥∥∥ N∑n=1

vnξn

∥∥∥pr

= EE∥∥∥ N∑n=1

vn (tn − tn−1)12 γn

∥∥∥pr

hp,r E∥∥∥( N∑

n=1

|vn|2 (tn − tn−1)) 1

2∥∥∥pr.

Remark 2.25. It follows from the definition of the Brownian motion that the

filtration (F βn )Nn=1, given by

F βn := σ

(β(tm), 1 ≤ m ≤ n

),

satisfies the properties needed in the previous lemma. See also Remark 1.19.

Chapter 3

Stochastic Integration

In this chapter we finally take up the study of stochastic integration in Lr spaces.

We first present a stochastic integral for functions f : [0, T ] → Lr(U), where

we assume that 1 ≤ r < ∞. While showing some of its properties, we will

observe that these integrals are Gaussian random variables. Using the results of

Chapter 2, we extend the first integral to Lr(U)-valued stochastic processes, now

assuming that 1 < r < ∞. In section 3.3 we will then see that this integral is a

time-continuous martingale. Unfortunately, due to the construction process, we

are bound to a strong integrability condition. To get rid of this condition, we will

apply a localization argument which allows the stochastic integral to be defined

for a much larger class of integrands.

3.1 The Wiener Integral

In this section we are going to define a stochastic integral for suitable functions

f : [0, T ] → Lr(U), 1 ≤ r < ∞, with respect to a Brownian motion(β(t)

)t∈[0,T ]

.

Such integrals are sometimes called Wiener integrals.

Definition 3.1. For an Lr(U)-valued step function of the form

f =N∑n=1

xn1[tn−1,tn) : [0, T ]→ Lr(U),

with x1, . . . , xN ∈ Lr(U) and 0 = t0 < . . . < tN = T , we define the random

variable∫ T

0f dβ : Ω→ Lr(U) by

∫ T

0

f dβ :=N∑n=1

xn(β(tn)− β(tn−1)

).

50 Stochastic Integration

Remark 3.2. By this definition, we have∫ T

0f dβ ∈ Lp(Ω;Lr(U)) for all 1 ≤

p <∞, satisfying

E∫ T

0

f dβ = 0

for each step function f .

In order to extend the stochastic integral to a broader class of functions, we make

the following observation.

Lemma 3.3 (Ito isomorphism for step functions). For all 1 ≤ p < ∞and all step functions f : [0, T ]→ Lr(U) we have(

E∥∥∥∫ T

0

f dβ∥∥∥pr

) 1p

hp,r

∥∥∥(∫ T

0

|f |2 dt) 1

2∥∥∥r. (3.1)

Proof. Let x1, . . . , xN ∈ Lr(U) and 0 = t0 < . . . < tN = T . Then, for f :=∑Nn=1 xn1[tn−1,tn), we have

∫ T

0

|f |2 dt =

∫ T

0

∣∣∣ N∑n=1

xn1[tn−1,tn)

∣∣∣2 dt =

∫ T

0

N∑n=1

|xn|21[tn−1,tn) dt

=N∑n=1

|xn|2(tn − tn−1),

where we used that the sets of the indicator functions are disjoint.

Now set γn := 1√tn−tn−1


), 1 ≤ n ≤ N . Then, by the definition of

the Brownian motion, (γn)Nn=1 is a sequence of independent standard Gaussian

variables. Hence, the Kahane inequality leads to(E∥∥∥∫ T

0

f dβ∥∥∥pr

) 1p

=

(E∥∥∥ N∑n=1


) ∥∥∥pr

) 1p

=

(E∥∥∥ N∑n=1

xn√tn − tn−1γn

∥∥∥pr

) 1p

hp,r

∥∥∥( N∑n=1

|xn|2(tn − tn−1)) 1

2∥∥∥r

=∥∥∥(∫ T

0

|f |2 dt) 1

2∥∥∥r.

3.1 The Wiener Integral 51

By computation, the norm of the space Lr(U ;L2([0, T ])) emerged in the previous

lemma, and in what follows, we will see that this space is the correct space of

integrands. It is for this reason that we will use the abbreviation

Lrγ,T := Lr(U ;L2([0, T ]))

for the rest of this thesis.

Lemma 3.4. The space of all step functions, given by

Dr,T :=f =

N∑n=1

xn1[tn−1,tn) : (xn)Nn=1 ⊆ Lr(U), 0 = t0 < . . . < tN = T,

is dense in Lrγ,T .

Proof. (1) We first show an auxiliary result. For this purpose we define

D1 :=f =

N∑n=1

gn1An : (gn)Nn=1 ⊆ L2([0, T ]), (An)Nn=1 ⊆ Σ with µ(An) <∞.

Now let ε > 0, and choose an arbitrary f1 =∑N

n=1 gn1An ∈ D1, where we may

assume that µ(An) > 0 for every n = 1, . . . , N . Since gn ∈ L2([0, T ]), there exists

a function

hn =Kn∑k=1

α(n)k 1

[t(n)k−1,t

(n)k )

with(α

(n)k

)Knk=1⊆ R, 0 = t

(n)0 < . . . < t

(n)Kn

= T , and ‖gn − hn‖L2([0,T ]) <ε

Nµ(An)1r

for each n = 1, . . . , N . We next define

f2 :=N∑n=1

hn1An =N∑n=1

Kn∑k=1

α(n)k 1

[t(n)k−1,t

(n)k )

1An ,

which is an element of Dr,T and satisfies ‖f1 − f2‖Lrγ,T < ε. Let us prove this.

Since(t(n)k

)n=1,...,N

k=0,...,Kn⊆ R, we can arrange the sequence in ascending order. Let

(ti)Mi=0 be the sequence sorted this way, with M :=

∑Nn=1Kn +N − 1. For every

set [ti−1, ti) and all n = 1, . . . , N , there exists exactly one ki,n ∈ 1, . . . , Kn with

[ti−1, ti) ⊆ [t(n)ki,n−1, t

(n)ki,n

). For i = 1, . . . ,M we define

xi :=N∑n=1

α(n)ki,n

1An .


By recalling that µ(An) < ∞ for all n = 1, . . . , N , we infer that xi ∈ Lr(U) for

each i = 1, . . . ,M , and we have

f2 =N∑n=1

Kn∑k=1

α(n)k 1

[t(n)k−1,t

(n)k )

1An =M∑i=1

xi1[ti−1,ti),

which means that f2 ∈ Dr,T . Moreover,

‖f1 − f2‖Lrγ,T =

(∫U

∥∥∥ N∑n=1

(gn − hn)1An

∥∥∥rL2([0,T ])

dµ

) 1r

≤N∑n=1

(∫U

1An‖gn − hn‖rL2([0,T ]) dµ) 1r

<N∑n=1

ε

Nµ(An)1r

(∫U

1An dµ) 1r

=N∑n=1

ε

N= ε.

(2) Now we return to the actual proof. Note that Dr,T ⊆ Lrγ,T . Let again ε > 0,

and choose an arbitrary f ∈ Lrγ,T . Observe that D1Lrγ,T = Lrγ,T (cf. Chapter A.1).

Therefore, there exists an h ∈ D1 with ‖f − h‖Lrγ,T <ε2. By (1) we can find a

g ∈ Dr,T with ‖h− g‖Lrγ,T <ε2, and this leads to

‖f − g‖Lrγ,T ≤ ‖f − h‖Lrγ,T + ‖h− g‖Lrγ,T < ε.

By Lemma 3.3, Lemma 3.4, and the open mapping theorem, we get the following

result.

Theorem 3.5 (Ito isomorphism). For every 1 ≤ p < ∞, the linear map-

ping

Iβ : Dr,T → Lp(Ω;Lr(U)), f 7→∫ T

0

f dβ

has a unique extension to a bounded operator

Iβ : Lrγ,T → Lp(Ω;Lr(U)),

which is an isomorphism onto its range and satisfies (3.1).

This motivates the next definition.


Definition 3.6. Let f ∈ Lrγ,T and (fn)∞n=1 ⊆ Dr,T be an approximating

sequence of f . Then we define the stochastic integral of f with respect to(β(t)

)t∈[0,T ]

by ∫ T

0

f dβ := Iβ(f) = limn→∞

∫ T

0

fn dβ,

where the convergence holds in Lp(Ω;Lr(U)) for each 1 ≤ p <∞.

Remark 3.7. (1) Lemma 3.4 guarantees that such an approximating sequence

always exists, and Theorem 3.5 ensures that∫ T

0f dβ ∈ Lp(Ω;Lr(U)) for each

1 ≤ p <∞ and every f ∈ Lrγ,T .

(2) The stochastic integral is well-defined in the sense that it is independent of

the approximating sequence.

(3) In general, the space Lrγ,T does not consist of functions f : [0, T ] → Lr(U),

even though the functions of Dr,T are dense in this space.

Remark 3.8. Taking a closer look on the case r = 2 and p = 2, we note that we

have equality in (3.1). This follows from Lemma 1.1 (observe that the constant

is 1 in this case) and Fubini’s theorem. Moreover, we have

〈h1, h2〉 =1

4

(‖h1 + h2‖2

2 − ‖h1 − h2‖22

)for all h1, h2 ∈ L2(U). Using this, Theorem 3.5, and Fubini’s theorem, we obtain

for all step functions f and g the formula

E⟨∫ T

0

f dβ,

∫ T

0

g dβ⟩

=1

4

(E∥∥∥∫ T

0

f + g dβ∥∥∥2

2− E

∥∥∥∫ T

0

f − g dβ∥∥∥2

2

)=

1

4

(∥∥∥(∫ T

0

|f + g|2 dt) 1

2∥∥∥2

2−∥∥∥(∫ T

0

|f − g|2 dt) 1

2∥∥∥2

2

)=

1

4

(∫U

∫ T

0

|f + g|2 dt dµ−∫U

∫ T

0

|f − g|2 dt dµ

)=

1

4

(∫ T

0

‖f + g‖22 dt−

∫ T

0

‖f − g‖22 dt

)=

∫ T

0

〈f, g〉 dt.

Next, we collect some properties of this stochastic integral.


Corollary 3.9. For every f ∈ Lrγ,T and 0 ≤ s ≤ t ≤ T we almost surely

have ∫ t

s

f dβ :=

∫ t

s

f |[s,t) dβ =

∫ T

0

1[s,t)f dβ.

Proof. (1) Let f ∈ Dr,T . Then there exists an N ∈ N, (xn)Nn=1 ⊆ Lr(U), and

0 = t0 < . . . < tN = T with f =∑N

n=1 xn1[tn−1,tn). Moreover, there exists a

k ∈ 1, . . . , N satisfying tk−1 ≤ t ≤ tk. Therefore,

1[0,t)f =k−1∑n=1

xn1[tn−1,tn) + xk1[tk−1,t) + 0 · 1[t,T ).

Clearly, 1[0,t)f ∈ Dr,t ∩Dr,T and 1[0,t)f = f |[0,t) on [0, t). We compute

∫ T

0

1[0,t)f dβ =k−1∑n=1


)+ xk

(β(t)− β(tk−1)

)=

∫ t

0

f dβ.

Now let f ∈ Lrγ,T and (fn)∞n=1 ⊆ Dr,T be an approximating sequence of f . Then

we have

limn→∞

‖1[0,t)fn − 1[0,t)f‖Lrγ,T = limn→∞

‖fn − f‖Lrγ,t ≤ limn→∞

‖fn − f‖Lrγ,T = 0,

which leads to∫ T

0

1[0,t)f dβ = limn→∞

∫ T

0

1[0,t)fn dβ = limn→∞

∫ t

0

fn dβ =

∫ t

0

f dβ in Lp(Ω;Lr(U)).

(2) Similarly as in (1), we obtain∫ T

s

f dβ =

∫ T

0

1[s,T )f dβ.

Therefore, we finally get∫ t

s

f dβ =

∫ t

0

1[s,t)f dβ =

∫ T

0

1[0,t)1[s,t)f dβ =

∫ T

0

1[s,t)f dβ.

Corollary 3.10. For every f ∈ Lrγ,T , the random variable∫ T

0f dβ is Gaus-

sian, especially E∫ T

0f dβ = 0.


Proof. (1) For a step function f =∑N

n=1 xn1[tn−1,tn) with x1, . . . , xN ∈ Lr(U)

and 0 = t0 < . . . < tN = T , we set

γn :=1√

tn − tn−1


)for each n = 1, . . . , N . Then, by the definition of the Brownian motion, (γn)Nn=1

is a sequence of independent standard Gaussian variables, and therefore∫ T

0

f dβ =N∑n=1


)=

N∑n=1

xn√tn − tn−1γn

is a Gaussian random variable.

(2) Now, let f ∈ Lrγ,T and (fn)∞n=1 ⊆ Dr,T be an approximating sequence of f .

Then, for every 1 ≤ p <∞, we have

limn→∞

∫ T

0

fn dβ =

∫ T

0

f dβ in Lp(Ω;Lr(U)).

Hence, Theorem 1.16 and Corollary 1.9 conclude the proof.

Remark 3.11. By the previous corollary and Theorem 1.16, the Lp convergence

in Definition 3.6 is equivalent to convergence in probability.

In what follows, it may be of interest under which conditions a function f : [0, T ]→Lr(U), which satisfies 〈f, g〉 ∈ L2([0, T ]) for all g ∈ Lr′(U), is already an element

of Lrγ,T , i.e., has a well-defined stochastic integral. For this purpose we need the

following proposition.

Proposition 3.12. Fix 1 ≤ p < ∞ and define, for n ∈ N0, the linear

operator An : Lp([0, T ];Lr(U))→ Lp([0, T ];Lr(U)) by

Anf :=2n∑j=1

1[(j−1)T

2n, jT2n

)xj,n,

where

xj,n :=2n

T

∫ jT2n

(j−1)T2n

f dt ∈ Lr(U).

Then we have limn→∞Anf=f in Lp([0, T ];Lr(U)) for all f ∈ Lp([0, T ];Lr(U)).


Proof. Let n ∈ N0 and f ∈ Lp([0, T ];Lr(U)). Since the dyadic intervals are

disjoint, we get

‖Anf‖pr =2n∑j=1

1[(j−1)T

2n, jT2n

)‖xj,n‖pr ≤

2n∑j=1

1[(j−1)T

2n, jT2n

)

(2n

T

∫ jT2n

(j−1)T2n

‖f‖r ds

)p.

Since 2n

Tds is a probability measure, and since x 7→ |x|p is a convex function,

Jensen’s inequality yield

‖Anf‖pr ≤2n∑j=1

1[(j−1)T

2n, jT2n

)

2n

T

∫ jT2n

(j−1)T2n

‖f‖pr ds.

Therefore, we obtain∫ T

0

‖Anf‖pr dt ≤2n∑j=1

∫ T

0

2n

T1

[(j−1)T

2n, jT2n

)dt

∫ jT2n

(j−1)T2n

‖f‖pr ds

=2n∑j=1

∫ jT2n

(j−1)T2n

‖f‖pr ds =

∫ T

0

‖f‖pr ds.

Hence, An is bounded and ‖An‖ ≤ 1.

We next want to show that dyadic step functions are dense in Lp([0, T ];Lr(U)).

Since continuous functions are dense in this space, it suffices to approximate them

by dyadic step functions. For this purpose, let f ∈ C([0, T ];Lr(U)). Then f is

uniformly continuous. Let ε > 0 be arbitrary. Then there exists a δ > 0 such

that |s− t| < δ implies ‖f(s)−f(t)‖r < ε for all s, t ∈ [0, T ]. For k ∈ N we define

a dyadic step function by fk := Akf . By choosing k so large such that T2k< δ,

we obtain similar as above∫ T

0

‖fk − f‖pr dt =

∫ T

0

2k∑j=1

1[(j−1)T

2k, jT2k

)(t)∥∥∥ 2k

T

∫ jT

2k

(j−1)T

2k

f(s) ds− f(t)∥∥∥pr

dt

≤2k∑j=1

∫ jT

2k

(j−1)T

2k

2k

T

∫ jT

2k

(j−1)T

2k

‖f(s)− f(t)‖pr ds dt

≤2k∑j=1

∫ jT

2k

(j−1)T

2k

εp dt = Tεp.

Since ε > 0 was arbitrary, this shows the desired density result.

Finally, for a dyadic step function f we have Anf = f for n large enough, and

the result now follows by the uniform boundedness of the operators (An)∞n=1.


Theorem 3.13. Fix 1 ≤ p < ∞. For a measurable function f : [0, T ] →Lr(U), which satisfies 〈f, g〉 ∈ L2([0, T ]) for all g ∈ Lr

′(U), the following

assertions are equivalent:

(1) f ∈ Lrγ,T ;

(2) there exists a random variable X : Ω → Lr(U) such that for all g ∈Lr′(U)

〈X, g〉 =

∫ T

0

〈f, g〉 dβ almost surely.

In this situation, we have X =∫ T

0f dβ almost surely.

Proof. (1) ⇒ (2): Let g ∈ Lr′(U) be arbitrary, and let (fn)∞n=1 ⊆ Dr,T be an

approximating sequence for f . Take X :=∫ T

0f dβ ∈ Lp(Ω;Lr(U)) as well as

Xn :=∫ T

0fn dβ. Then limn→∞Xn = X in Lp(Ω;Lr(U)), and for each n ∈ N and

appropriate sequences 0 = t(n)1 < . . . < t

(n)Nn

= T and (x(n)k )Nnk=1 ⊆ Lr(U), almost

surely we have

〈Xn, g〉 =Nn∑k=1

〈x(n)k , g〉

(β(t

(n)k )− β(t

(n)k−1)

)=

∫ T

0

〈fn, g〉 dβ.

This implies

limn→∞

∫ T

0

〈fn, g〉 dβ = limn→∞〈Xn, g〉 = 〈X, g〉 in Lp(Ω). (4)

Since (U,Σ, µ) is a σ-finite measure space, U is a countable union of disjoint sets

Uk ∈ Σ, k ∈ N, of finite measure. Take U (K) :=⋃Kk=1 Uk. By Holder’s inequality

and Fubini’s theorem we obtain for each fixed K ∈ N

limn→∞

1U(K)fn = 1U(K)f in L1([0, T ]× U (K)),

since limn→∞ fn = f in Lrγ,T . Moreover, there exists an appropriate subsequence

and a set N (K) ⊆ [0, T ]× U (K) of measure 0, such that

limj→∞

1U(K)(u)fnj(t, u) = 1U(K)(u)f(t, u)

and hence

limj→∞

1U(K)(u)fnj(t, u)g(u) = 1U(K)(u)f(t, u)g(u)

for all (t, u) ∈([0, T ]× U (K)

)\N (K).


We note that 1U(K)f(t) ∈ Lr(U), and thus 1U(K)f(t)g ∈ L1(U) for each t ∈ [0, T ].

Therefore, by the dominated convergence theorem, we get for almost all t ∈ [0, T ]

limj→∞〈1U(K)fnj(t), g〉 = lim

j→∞

∫U

1U(K)fnj(t)g dµ =

∫U

1U(K)f(t)g dµ

= 〈1U(K)f(t), g〉.

Since 1U(K)g ∈ Lr′(U), we have 〈1U(K)f, g〉 = 〈f,1U(K)g〉 ∈ L2([0, T ]), by assump-

tion. Now, again the dominated convergence theorem yield

limj→∞〈fnj ,1U(K)g〉 = lim

j→∞〈1U(K)fnj , g〉 = 〈1U(K)f, g〉 = 〈f,1U(K)g〉 in L2([0, T ]).

And together with (4) we infer that∫ T

0

〈f,1U(K)g〉 dβ = limj→∞

∫ T

0

〈fnj ,1U(K)g〉 dβ = 〈X,1U(K)g〉 in Lp(Ω).

Once more by the dominated convergence theorem we have

limK→∞

〈X,1U(K)g〉 = 〈X, g〉 in Lp(Ω)

and

limK→∞

〈f,1U(K)g〉 = 〈f, g〉 in L2([0, T ]),

and this finally leads to

〈X, g〉 = limK→∞

〈X,1U(K)g〉 = limK→∞

∫ T

0

〈f,1U(K)g〉 dβ =

∫ T

0

〈f, g〉 dβ in Lp(Ω).

(2) ⇒ (1): For each k ∈ N we set f (k)(t) := 1‖f(t)‖r≤k(t)f(t), t ∈ [0, T ]. Note

that f (k) ∈ L2([0, T ];Lr(U)). Moreover, since limn→∞〈f (n)(t), g〉 = 〈f(t), g〉 for

all t ∈ [0, T ] and 〈f, g〉 ∈ L2([0, T ]) for all g ∈ Lr′(U), we obtain

limn→∞〈f (n), g〉 = 〈f, g〉 in L2([0, T ])

by the dominated convergence theorem. Next, define fn := Anf(n) for n ∈ N, and

with An being the operator from Proposition 3.12. Since f (n) ∈ L2([0, T ];Lr(U)),

we have for each g ∈ Lr′(U) and all j = 1, . . . , 2n

⟨∫ jT2n

(j−1)T2n

f (n)(t) dt, g⟩

=

∫ jT2n

(j−1)T2n

〈f (n), g〉(t) dt,

and this leads to

〈fn, g〉 =2n∑j=1

1[(j−1)T

2n, jT2n

)

2n

T

∫ jT2n

(j−1)T2n

〈f (n), g〉(t) dt = An〈f (n), g〉.


Therefore, by the properties of An, we have

limn→∞

‖〈fn, g〉 − 〈f, g〉‖L2([0,T ])

≤ limn→∞

(‖〈fn, g〉 − An〈f, g〉‖L2([0,T ]) + ‖An〈f, g〉 − 〈f, g〉‖L2([0,T ])

)(?)

≤ limn→∞

(‖〈f (n), g〉 − 〈f, g〉‖L2([0,T ]) + ‖An〈f, g〉 − 〈f, g〉‖L2([0,T ])

)= 0.

Next, we define Xn :=∫ T

0fn − fn−1 dβ for n ∈ N and with f0 := 0. Then we get

〈Xn, h〉 =

∫ T

0

〈fn − fn−1, h〉 dβ

for each n ∈ N and every h ∈ Lr′(U). We now want to show that the random

variables Xn are independent. By the linearity of the stochastic integral, the

random variables Xn are jointly Gaussian. Therefore, by Proposition A.16, it

suffices to check that E〈Xm, h1〉〈Xn, h2〉 = 0 for m 6= n and h1, h2 ∈ Lr′(U).

Together with Remark 3.8 we obtain

E〈Xm, h1〉〈Xn, h2〉 =

∫ T

0

〈fm − fm−1, h1〉〈fn − fn−1, h2〉 dt.

For j = 1, . . . , 2m and k = 1, . . . , 2n we define

xj,m :=2m

T

∫ jT2m

(j−1)T2m

〈f (m), h1〉 dt, yk,n :=2n

T

∫ kT2n

(k−1)T2n

〈f (n), h2〉 dt.

Let m > n. Then we obtain

E〈Xm, h1〉〈Xn, h2〉 =

∫ T

0

( 2m∑j=1

1[(j−1)T

2m, jT2m

)xj,m −

2m−1∑j=1

1[(j−1)T

2m−1 ,jT

2m−1 )xj,m−1

)

·( 2n∑k=1

1[(k−1)T

2n, kT2n

)yk,n −

2n−1∑k=1

1[(k−1)T

2n−1 , kT2n−1 )

yk,n−1

)dt

=T

2m

2n∑k=1

( k2m−n∑l=1+(k−1)2m−n

xl,m

)yk,n −

T

2m−1

2n∑k=1

( k2m−n−1∑l=1+(k−1)2m−n−1

xl,m−1

)yk,n

− T

2m

2n−1∑k=1

( k2m−n+1∑l=1+(k−1)2m−n+1

xl,m

)yk,n−1 +

T

2m−1

2n−1∑k=1

( k2m−n∑l=1+(k−1)2m−n

xl,m−1

)yk,n−1

=2n∑k=1

( T2m

k2m−n∑l=1+(k−1)2m−n

xl,m −T

2m−1

k2m−n−1∑l=1+(k−1)2m−n−1

xl,m−1

)yk,n

−2n−1∑k=1

( T2m

k2m−n+1∑l=1+(k−1)2m−n+1

xl,m −T

2m−1

k2m−n∑l=1+(k−1)2m−n

xl,m−1

)yk,n−1 = 0,


since for each k we have

T

2m

k2m−n∑l=1+(k−1)2m−n

xl,m =

∫ kT2n

(k−1)T2n

〈f (m), h1〉 dt =T

2m−1

k2m−n−1∑l=1+(k−1)2m−n−1

xl,m−1,

T

2m

k2m−n+1∑l=1+(k−1)2m−n+1

xl,m =

∫ kT2n−1

(k−1)T

2n−1

〈f (m), h1〉 dt =T

2m−1

k2m−n∑l=1+(k−1)2m−n

xl,m−1.

Now take SN :=∑N

n=1Xn =∫ T

0fN dβ. Then, by (2), Theorem 3.5, and (?), we

obtain

limN→∞

E∣∣〈X, g〉 − 〈SN , g〉∣∣2 = lim

N→∞E(∫ T

0

〈f, g〉 dβ −∫ T

0

〈fN , g〉 dβ)2

= limN→∞

E(∫ T

0

〈f − fN , g〉 dβ)2

= limN→∞

∫ T

0

〈f − fN , g〉2 dt = 0

for all g ∈ Lr′(U). Then, by Proposition A.17, X is a Gaussian random variable,

and by Fernique’s theorem we deduce that X ∈ Lp(Ω;Lr(U)). Thus, applying

Proposition 1.13 leads to

limN→∞

∫ T

0

fN dβ = X in Lp(Ω;Lr(U)).

By this convergence and Theorem 3.5, (fn)∞n=1 is Cauchy in Lrγ,T with limit f . As

in the first part of this proof, we can show that

limj→∞

1U(N)(u)fnj(t, u) = 1U(N)(u)f(t, u)

for almost all (t, u) ∈ [0, T ]× U (N) and any fixed N ∈ N, with U (N) ∈ Σ defined

as above. And similarly, we can show that

limk→∞

1U(N)(u)fnjk (t, u) = 1U(N)(u)f(t, u)

for almost all (t, u) ∈ [0, T ] × U (N) and all N ∈ N, since limj→∞ fnj = f in

L2([0, T ];Lr(U)). Therefore, we have f = f ∈ Lrγ,T . This also shows that

X =

∫ T

0

f dβ =

∫ T

0

f dβ almost surely.

3.2 The Ito Integral 61

3.2 The Ito Integral

In the previous section we were introduced to a stochastic integral of functions

f : [0, T ]→ Lr(U). Next, we are going to define a stochastic integral for processes

f : [0, T ]× Ω → Lr(U), where we now assume that 1 < r < ∞. As it turns out,

the Decoupling theorem will play a central role.

Definition 3.14. A function f : [0, T ]×Ω→ Lr(U) is said to be an adapted

step process with respect to a filtration F = (Ft)t∈[0,T ] if it is of the form

f(t, ω) =N∑n=1

1[tn−1,tn)(t)Kn∑k=1

x(n)k 1

A(n)k

(ω), (3.2)

where 0 = t0 < . . . < tN = T and, for all n = 1, . . . , N , x(n)1 , . . . , x

(n)Kn∈ Lr(U)

and A(n)1 , . . . , A

(n)Kn∈ Ftn−1 .

For the rest of this section we assume that(β(t)

)t∈[0,T ]

is a Brownian motion

which is adapted to the filtration F in the sense that for all 0 ≤ s < t ≤ T

(1) β(t) is Ft-measurable and

(2) β(t)− β(s) is independent of Fs.

Definition 3.15. The stochastic integral with respect to(β(t)

)t∈[0,T ]

of an

adapted step process f of the form (3.2) is defined by

∫ T

0

f dβ :=N∑n=1

Kn∑k=1

x(n)k 1

A(n)k


).

Remark 3.16. It is straight forward to check that this definition does not de-

pend on the particular representation of f . Also, if we choose A(n)k = Ω for all

n = 1, . . . , N and each k = 1, . . . , Kn, this integral coincides with the Wiener

integral. From this point of view, the Ito integral is an extension of the integral

given in section 3.1.

Additionally we have the following properties.

Proposition 3.17. For every adapted step process f we have∫ T

0f dβ ∈

Lp(Ω,FT ;Lr(U)) for all 1 ≤ p <∞ satisfying E∫ T

0f dβ = 0.


Proof. Clearly, the random variable∫ T

0f dβ is FT -measurable. By Remark 1.8

we have(E|β(t)−β(s)|p

) 1p <∞ for every fixed 0 ≤ s < t ≤ T and all 1 ≤ p <∞.

Hence,(E∥∥∥∫ T

0

f dβ∥∥∥pr

) 1p

=

(E∥∥∥ N∑n=1

Kn∑k=1

x(n)k 1

A(n)k


) ∥∥∥pr

) 1p

≤N∑n=1

Kn∑k=1

‖x(n)k ‖r

(E∣∣β(tn)− β(tn−1)

∣∣p) 1p<∞.

In addition,

E(1A

(n)k


))= E

(E[1A

(n)k


)∣∣Ftn−1

])= E

(1A

(n)kE[(β(tn)− β(tn−1)

)∣∣Ftn−1

])= E

(1A

(n)kE(β(tn)− β(tn−1)

))= 0,

using that 1A

(n)k

is Ftn−1-measurable and the fact that β(tn) − β(tn−1) is inde-

pendent of Ftn−1 . Therefore, by linearity,

E∫ T

0

f dβ =N∑n=1

Kn∑k=1

x(n)k E

(1A

(n)k


))= 0.

Similar to Section 3.1, we want to give an estimate for the Lp(Ω;Lr(U))-norm of∫ T0f dβ. Here we are going to use results of Section 2.3, and therefore, we have

to confine ourselves to the cases 1 < p <∞.

Lemma 3.18 (Ito isomorphism for adapted step processes). For 1 <

p <∞ and every adapted step process f : [0, T ]× Ω→ Lr(U), we have

E∥∥∥∫ T

0

f dβ∥∥∥prhp,r E

∥∥∥(∫ T

0

|f |2 dt) 1

2∥∥∥pr. (3.3)

Proof. Let f be an adapted step process of the form (3.2). Then we define for

n = 1, . . . , N

vn :=Kn∑k=1

x(n)k 1

A(n)k

and Fn := Ftn .

So, (vn)Nn=1 is predictable with respect to the filtration (Fn)Nn=1 since every A(n)k

is Fn−1-measurable. Additionally, by the general assumption in this section,

(Fn)Nn=1 fulfills the conditions of Corollary 2.24.


Moreover,∫ T

0

|f |2 dt =

∫ T

0

∣∣∣ N∑n=1

vn1[tn−1,tn)

∣∣∣2 dt =N∑n=1

∫ T

0

|vn|21[tn−1,tn) dt

=N∑n=1

|vn|2(tn − tn−1).

Finally, by applying Corollary 2.24, we obtain

E∥∥∥∫ T

0

f dβ∥∥∥pr

= E∥∥∥ N∑n=1


) ∥∥∥pr

hp,r E∥∥∥( N∑

n=1

|vn|2(tn − tn−1)) 1

2∥∥∥pr

= E∥∥∥(∫ T

0

|f |2 dt) 1

2∥∥∥pr.

Definition 3.19. A random variable f ∈ Lp(Ω;Lrγ,T ) is called an adapted

Lp process with respect to a filtration F if

1[0,t)f ∈ Lp(Ω,Ft;Lrγ,t) for all t ∈ [0, T ],

which means that f has a representative f : [0, T ] × Ω × U → R such that

1[0,t)f is B([0, t))⊗Ft ⊗ Σ-measurable.

We denote by LpF(Ω;Lrγ,T ) the closed subspace in Lp(Ω;Lrγ,T ) of all F-adapted

elements.

Remark 3.20. Functions with the property given in the definition above are

often called progressively measurable, whereas functions f : [0, T ] × Ω × U → Rare usually called adapted if f(t) is Ft⊗Σ-measurable. But since the first implies

the latter, the above definition is still meaningful.

By Definition 3.19, we get the following density result.

Lemma 3.21. For every 1 ≤ p <∞, the space of all adapted step processes

with respect to a filtration F is dense in LpF(Ω;Lrγ,T ).

Proof. Note that the space of adapted step processes is a subset of LpF(Ω;Lrγ,T ).


(1) We first assume that µ(U) < ∞. Taking q := maxr, p, 2, we observe that

LqF(Ω;Lq(U ;Lq([0, T ]))) is dense in LpF(Ω;Lrγ,T ). Therefore it suffices to approxi-

mate functions in the first space. By Fubini’s theorem we have

Lq(Ω;Lq(U ;Lq([0, T ]))) ' Lq([0, T ];Lq(Ω;Lq(U))) =: E.

Let f ∈ E be an adapted process and ε > 0. Choose δ > 0 such that ‖fδ−f‖E < ε

for fδ := 1[0,T−δ]f(· − δ) ∈ E, and define for n ∈ N

fn := Anfδ =2n∑j=1

1[tj−1,n,tj,n)vj,n,

where

tj,n :=jT

2n, vj,n :=

2n

T

∫ tj,n

tj−1,n

fδ dt ∈ Lq(Ω;Lq(U)),

and An is the operator from Proposition 3.12. Now choose N ∈ N so large such

that ‖fN − fδ‖E < ε and T2N

< δ. Then we have ‖f − fN‖E < 2ε and

vj,N =2N

T

∫ tj,N

tj−1,N

fδ dt =2N

T

∫ tj,N−δ

tj−1,N−δf dt.

Since tj,N − δ < tj,N − T2N

= tj−1,N and f is an adapted process, we infer that f is

Ftj−1,N-measurable on [0, tj,N−δ). This implies that vj,N ∈ Lq(Ω,Ftj−1,N

;Lq(U)).

Finally, we can find for each j = 1, . . . , 2N a simple function satisfying

∥∥∥ vj,N − Nj∑k=1

x(j)k 1

A(j)k

∥∥∥Lq(Ω;Lq(U))

<ε

2N2Nq

T1q

with A(j)k ∈ Ftj−1,N

. Then,

g :=2N∑j=1

1[tj−1,N ,tj,N )

Nj∑k=1

x(j)k 1

A(j)k

is an adapted step process with respect to F, and

‖fN − g‖E =

(∫ T

0

‖fN − g‖qLq(Ω;Lq(U)) dt

) 1q

≤2N∑j=1

(∫ T

0

1[tj−1,N ,tj,N )

∥∥∥ vj,N − Nj∑k=1

x(j)k 1

A(j)k

∥∥∥qLq(Ω;Lq(U))

dt

) 1q

<2N∑j=1

ε

2N

(∫ T

0

1[tj−1,N ,tj,N )2N

Tdt

) 1q

= ε.


Hence, ‖f − g‖E < 3ε. By Holder’s inequality we have

‖ · ‖Lp(Ω;Lrγ,T ) . ‖ · ‖Lq(Ω;Lq(U ;Lq([0,T ]))).

Therefore,

‖f − g‖Lp(Ω;Lrγ,T ) . ‖f − g‖Lq(Ω;Lq(U ;Lq([0,T ]))) = ‖f − g‖E < 3ε.

(2) Now assume that (U,Σ, µ) is a σ-finite measure space, and let ε > 0. Then

U =⋃∞n=1 Un for sets Un ∈ Σ with µ(Un) < ∞. Given f ∈ LpF(Ω;Lrγ,T ) and

N ∈ N, we define

fN := 1⋃Nn=1 Un

f.

Since f ∈ Lp(Ω;Lrγ,T ) and limN→∞ fN(ω, u) = f(ω, u) in L2([0, T ]) for all ω ∈Ω and u ∈ U , the dominated convergence theorem yield limN→∞ fN = f in

Lp(Ω;Lrγ,T ). Thus, we can choose an N ∈ N such that ‖f − fN‖Lp(Ω;Lrγ,T ) < ε.

Since fN has support of finite measure, step (1) donates an adapted step process

g which satisfies ‖fN − g‖Lp(Ω;Lrγ,T ) < ε. This finally yields

‖f − g‖Lp(Ω;Lrγ,T ) < 2ε.

By Lemma 3.18 and Lemma 3.21 we get the next result.

Theorem 3.22 (Ito isomorphism for adapted Lp processes). For ev-

ery 1 < p <∞ the stochastic integral extends uniquely to a bounded operator

IβF : LpF(Ω;Lrγ,T )→ Lp(Ω,FT ;Lr(U)),

which is an isomorphism onto its range and satisfies (3.3).

As in Section 3.1, we can now define a stochastic integral for adapted Lp processes.

Definition 3.23. Fix 1 < p < ∞. Let f ∈ LpF(Ω;Lrγ,T ) and (fn)∞n=1 be

an approximating sequence of adapted step processes. Then we define the

stochastic integral of f by∫ T

0

f dβ := IβF (f) = limn→∞

∫ T

0

fn dβ,

where convergence holds in Lp(Ω,FT ;Lr(U)).


Remark 3.24. Similar to Remark 3.7, we have∫ T

0f dβ ∈ Lp(Ω;Lr(U)) for all

f ∈ LpF(Ω;Lrγ,T ), and it is straight forward to check that the stochastic integral

does not depend on the approximating sequence of f . Further, note that the space

LpF(Ω;Lrγ,T ) does generally not consist of functions f : [0, T ]× Ω→ Lr(U).

We now have the following equivalent to Corollary 3.9.

Corollary 3.25. For 1 < p < ∞, every f ∈ LpF(Ω;Lrγ,T ), and s, t ∈ [0, T ],

we have ∫ t

s

f dβ :=

∫ t

s

f |[s,t) dβ =

∫ T

0

1[s,t)f dβ.

Proof. This assertion is proved in a very similar way as Corollary 3.9. By re-

placing the sequence (xn)Nn=1 ⊆ Lr(U) with a sequence (∑Kn

k=1 x(n)k 1

A(n)k

)Nn=1 ⊆Lp(Ω;Lr(U)), the statement follows in the same way for adapted step processes

as for step functions. And once again it follows by an approximation argument

that the result holds for any adapted Lp process.

If we choose the augmented Brownian filtration Fβ := (F βt )t∈[0,T ] (i.e., F β

0 con-

tains all P-null sets), we can get some interesting results. Recall that Fβ fulfills

the conditions we assumed above. See also Remark 1.19 and Remark 2.25.

Lemma 3.26. Let 1 < p < ∞ and ξ ∈ Lp(Ω,F βT ). Then there exists a

unique φ ∈ LpFβ(Ω;L2([0, T ])) such that

ξ = Eξ +

∫ T

0

φ dβ.

Proof. Let (ξn)∞n=1 be a sequence of simple functions converging to ξ in Lp(Ω,F βT ).

For each n ∈ N we have ξn ∈ L2(Ω,F βT ). Thus, by the classical representation

theorem (cf. [14, Theorem 12.2]), we obtain a φn ∈ L2Fβ(Ω;L2([0, T ])) which

satisfies

ξn = Eξn +

∫ T

0

φn dβ.

Since limn→∞ ξn = ξ in Lp(Ω,F βT ), we have limn→∞ Eξn = Eξ. By Theorem 3.22

we infer that the sequence (φn)∞n=1 is Cauchy in LpFβ(Ω;L2([0, T ])). Hence, there

exists a unique limit φ ∈ LpFβ(Ω;L2([0, T ])) with the property

ξ − Eξ = limn→∞

ξn − Eξn = limn→∞

∫ T

0

φn dβ =

∫ T

0

φ dβ.


Theorem 3.27 (Representation theorem). Let 1 < p < ∞ and X ∈Lp(Ω,F β

T ;Lr(U)). Then there exists a unique f ∈ LpFβ(Ω;Lrγ,T ) such that

X = EX +

∫ T

0

f dβ.

Proof. There exists a sequence (Xn)∞n=1 of simple F βT -measurable random vari-

ables with limn→∞Xn = X in Lp(Ω;Lr(U)) (cf. Chapter A.1). For each n we

assume that Xn =∑Kn

k=1 1A(n)kx

(n)k , where A

(n)k ∈ F β

T and x(n)k ∈ Lr(U) for all k =

1, . . . , Kn. By Lemma 3.26, there exist unique processes φ(n)k ∈ L

pFβ(Ω;L2([0, T ]))

such that

1A

(n)k

= E1A

(n)k

+

∫ T

0

φ(n)k dβ.

Now define fn :=∑Kn

k=1 φ(n)k x

(n)k . Since

(E∥∥∥(∫ T

0

|fn|2 dt) 1

2∥∥∥pr

) 1p

=

(E∥∥∥(∫ T

0

∣∣∣ Kn∑k=1

φ(n)k x

(n)k

∣∣∣2 dt) 1

2∥∥∥pr

) 1p

≤Kn∑k=1

(E(∫ T

0

|φ(n)k |

2 dt) p

2‖x(n)k ‖

pr

) 1p

=Kn∑k=1

‖φ(n)k ‖Lp(Ω;L2([0,T ]))‖x(n)

k ‖r <∞

and every φ(n)k belongs to LpFβ(Ω;L2([0, T ])), we infer that fn ∈ LpFβ(Ω;Lrγ,T ).

Moreover,

Xn =Kn∑k=1

1A

(n)kx

(n)k =

Kn∑k=1

(E1

A(n)k

+

∫ T

0

φ(n)k dβ

)x

(n)k

= EKn∑k=1

1A

(n)kx

(n)k +

∫ T

0

Kn∑k=1

φ(n)k x

(n)k dβ

= EXn +

∫ T

0

fn dβ.

Since limn→∞Xn = X in Lp(Ω;Lr(U)), it holds that the sequence(∫ T

0fn dβ

)∞n=1

converges in Lp(Ω;Lr(U)), and therefore, the isomorphism IβFβ of Theorem 3.22

implies that the sequence (fn)∞n=1 is Cauchy in LpFβ(Ω;Lrγ,T ). Hence, there exists

a unique limit f ∈ LpFβ(Ω;Lrγ,T ), which satisfies∫ T

0

f dβ = limn→∞

∫ T

0

fn dβ = limn→∞

Xn − EXn = X − EX.


As a consequence of Theorem 3.22 and Theorem 3.27, we get the following corol-

lary.

Corollary 3.28. For 1 < p < ∞, the stochastic integral defines an isomor-

phism of Banach spaces

IβFβ : LpFβ(Ω;Lrγ,T ) ' Lp0(Ω,F βT ;Lr(U)),

where Lp0(Ω,F βT ;Lr(U)) is the closed subspace of Lp(Ω,F β

T ;Lr(U)) consisting

of all elements with mean 0.

Returning to a general filtration F, which satisfies the properties stated after

Definition 3.14, we finally extend Theorem 3.13.

Theorem 3.29. Fix 1 < p < ∞. For a measurable and adapted function

f : [0, T ] × Ω → Lr(U), which satisfies 〈f, g〉 ∈ Lp(Ω;L2([0, T ])) for all g ∈Lr′(U), the following assertions are equivalent:

(1) f ∈ LpF(Ω;Lrγ,T );

(2) there exists a random variable X ∈ Lp(Ω;Lr(U)) such that for all

g ∈ Lr′(U)

〈X, g〉 =

∫ T

0

〈f, g〉 dβ almost surely.

In this situation, we have X =∫ T


Proof. (1) ⇒ (2): This can be proved in the same way as the corresponding

implication of Theorem 3.13.

(2) ⇒ (1): For n ∈ N, let Pn : Lr(U) → Lr(U) be the operator from Lemma

A.26. Let h ∈ Lr(U), and recall that

limn→∞

Pnh = h in Lr(U),

and that Pn has the representation

(Pnh)(u) =Nn∑j=1

1

µ(U(n)j )

∫U

(n)j

h dµ 1U

(n)j

(u)

for suitable disjoint sets U(n)j ⊆ U , j = 1, . . . , Nn.


Hence, by Fubini’s theorem, we get for all g ∈ Lr′(U)

〈Pnh, g〉 =

∫U

( Nn∑j=1

1

µ(U(n)j )

∫U

(n)j

h dµ 1U

(n)j

)g dµ

=

∫U

h( Nn∑j=1

1

µ(U(n)j )

∫U

(n)j

g dµ 1U

(n)j

)dµ = 〈h, Png〉.

Now take fn(t, ω) := Pnf(t, ω) for (t, ω) ∈ [0, T ] × Ω. Since 〈fn, g〉 = 〈f, Png〉 ∈LpF(Ω;L2([0, T ])) for all g ∈ Lr′(U), we obtain for each k = 1, . . . , Nn

(t, ω) 7→∫U

(n)k

f(t, ω) dµ =⟨fn(t, ω), µ(U

(n)k )1

U(n)k

⟩∈ LpF(Ω;L2([0, T ])),

where the equality follows from the fact that the sets U(n)1 , . . . , U

(n)Nn

are disjoint.

Thus, we get

E∥∥∥∫

U(n)k

f dµ 1U

(n)k

∥∥∥pLrγ,T

= µ(U(n)k )

prE∥∥∥∫

U(n)k

f dµ∥∥∥pL2([0,T ])

<∞.

And therefore, we have

fn =Nn∑j=1

1

µ(U(n)j )

∫U

(n)j

f dµ 1U

(n)j∈ LpF(Ω;Lrγ,T )

for each n ∈ N. Hence, almost surely and for an arbitrary g ∈ Lr′(U) we have

the well-defined equalities∫ T

0

〈fn, g〉 dβ =Nn∑j=1

1

µ(U(n)j )

∫ T

0

∫U

(n)j

f dµ dβ 〈1U

(n)j, g〉

=⟨∫ T

0

Nn∑j=1

1

µ(U(n)j )

∫U

(n)j

f dµ 1U

(n)j

dβ, g⟩

=⟨∫ T

0

fn dβ, g⟩.

By this estimate and (2), we obtain

〈PnX, g〉 = 〈X,Png〉 =

∫ T

0

〈f, Png〉 dβ =

∫ T

0

〈fn, g〉 dβ =⟨∫ T

0

fn dβ, g⟩

almost surely and for all g ∈ Lr′(U). By Corollary A.8, this leads to

PnX =

∫ T

0

fn dβ almost surely. (4)


Next, by the properties of the operators (Pn)∞n=1 we mentioned above, almost

surely we have limn→∞ PnX = X in Lr(U), and since X ∈ Lp(Ω;Lr(U)), the

dominated convergence theorem yield

limn→∞

PnX = X in Lp(Ω;Lr(U)).

Looking at (4), this convergence and Theorem 3.22 imply that (fn)∞n=1 is Cauchy

in LpF(Ω;Lrγ,T ), and thus has a limit f . Similar as in the proof of Theorem 3.13,

we can show that

limj→∞

1U(N)(u)fnj(t, ω, u) = 1U(N)(u)f(t, ω, u)

for almost all (t, ω, u) ∈ [0, T ]× Ω× U , and moreover we have

limk→∞

1U(N)(u)fnjk (t, ω, u) = 1U(N)(u)f(t, ω, u)

for almost all (t, ω, u) ∈ [0, T ]×Ω×U , since for all (t, ω) we have limn→∞ fn = f

in Lr(U). From this we infer that f = f ∈ LpF(Ω;Lrγ,T ) and this also shows that

X =∫ T


3.3 The Integral Process

In this short section we want to investigate the properties of the integral process

t 7→∫ t

0

f dβ, t ∈ [0, T ],

where f ∈ LpF(Ω;Lrγ,T ) and F = (Ft)t∈[0,T ] is a filtration satisfying the conditions

we posted in the previous section. Here, we again assume that 1 < r <∞ is fixed.

Proposition 3.30. For each 1 < p <∞ and every f ∈ LpF(Ω;Lrγ,T ), the inte-

gral process(∫ t

0f dβ

)t∈[0,T ]

is an Lp martingale with respect to the filtration

F.

Proof. We first assume that f is an adapted step process of the form (3.2). Then,

by Proposition 3.17,∫ t

0f dβ is an Ft-measurable and integrable random variable

for every t ∈ [0, T ]. Now define vn :=∑Kn

k=1 x(n)k 1

A(n)k

, and fix 0 ≤ s < t ≤ T .

Then there exists a k ∈ 1, . . . , N such that tk ≤ s < tk+1, and we can write

f =k∑

n=1

vn1[tn−1,tn) + vk+1(1[tk,s) + 1[s,tk+1)) +N∑

n=k+2

vn1[tn−1,tn).

3.3 The Integral Process 71

For each 0 ≤ τ1 < τ2 ≤ T , we obtain

E[β(τ2)− β(τ1)|Fτ1 ] = E(β(τ2)− β(τ1)

)= 0,

since β(τ2)− β(τ1) is independent of Fτ1 by the properties of the filtration F we

assumed above. Additionally, vk+1 is Fs-measurable since tk ≤ s. Therefore, we

have

E[∫ t

0

f dβ∣∣Fs

]=

k∑n=1

E[vn(β(tn)− β(tn−1)

)∣∣Fs

]+ E

[vk+1

(β(s)− β(tk)

)∣∣Fs

]+ E

[vk+1

(β(tk+1)− β(s)

)∣∣Fs

]+

N∑n=k+2

E[vn(β(tn)− β(tn−1)

)∣∣Fs

]=

k∑n=1


)+ vk+1

(β(s)− β(tk)

)+ vk+1E[β(tk+1)− β(s)|Fs]

+N∑

n=k+2

E[E[vn(β(tn)− β(tn−1)

)∣∣Ftn−1

]∣∣∣Fs

]=

∫ s

0

f dβ +N∑

n=k+2

E[vnE

[(β(tn)− β(tn−1)

)∣∣Ftn−1

]∣∣∣Fs

]=

∫ s

0

f dβ,

using that the sequence (vn)Nn=1 is predictable with respect to the filtration

(Ftn)Nn=1.

Now, let f ∈ LpF(Ω;Lrγ,T ), and let (fn)∞n=1 be an approximating sequence of

adapted step processes for f . Since

limn→∞

1[0,t)fn = 1[0,t)f in LpF(Ω;Lrγ,t),

Theorem 3.22 and Corollary 3.25 imply that

limn→∞

∫ t

0

fn dβ =

∫ t

0

f dβ in Lp(Ω,Ft;Lr(U))

for each t ∈ [0, T ], which also yield that∫ t

0f dβ is an Ft-measurable and in-

tegrable random variable. Observe next that for each sub-σ-Algebra G of A

the operator E[ · |G ] : Lp(Ω;Lr(U)) → Lp(Ω,G ;Lr(U)) is continuous. Thus, we

obtain for 0 ≤ s < t ≤ T

E[∫ t

0

f dβ∣∣Fs

]= lim

n→∞E[∫ t

0

fn dβ∣∣Fs

]= lim

n→∞

∫ s

0

fn dβ =

∫ s

0

f dβ

in Lp(Ω,Fs;Lr(U)).


Theorem 3.31. For 1 < p < ∞ and f ∈ LpF(Ω;Lrγ,T ), the integral process(∫ t0f dβ

)t∈[0,T ]

has a continuous version satisfying the maximal inequality

E∥∥∥ supt∈[0,T ]

∣∣∣ ∫ t

0

f dβ∣∣∣ ∥∥∥p

r.p,r E

∥∥∥∫ T

0

f dβ∥∥∥pr.

Proof. Let (fn)∞n=1 be an approximating sequence of adapted step processes for

f , and set Xn(t) :=∫ t

0fn dβ. Then, by the definition of the stochastic integral for

adapted step processes and the properties of the Brownian motion (cf. Remark

1.23), Xn has a continuous version Xn ∈ Lp(Ω;Lr(U ;C([0, T ]))). By Proposition

3.30 and the Strong Doob inequality, we obtain for every choice of 0 = t1 < . . . <

tN = T

E∥∥ N

supj=1|Xn(tj)− Xm(tj)|

∥∥pr.p,r E‖Xn(T )−Xm(T )‖pr.

Thus, by continuity and Fatou’s lemma, we obtain

E∥∥ supt∈[0,T ]

|Xn(t)− Xm(t)|∥∥pr.p,r E‖Xn(T )−Xm(T )‖pr,

which shows that the sequence (Xn)∞n=1 is Cauchy in Lp(Ω;Lr(U ;C([0, T ]))),

and therefore has a limit X ∈ Lp(Ω;Lr(U ;C([0, T ]))) ⊆ C([0, T ];Lp(Ω;Lr(U))).

Since for all t ∈ [0, T ] we have limn→∞Xn(t) =∫ t

0f dβ and limn→∞ Xn(t) = X(t)

both in Lp(Ω;Lr(U)), X defines a continuous version of the integral process.

Finally, the maximal inequality follows from the Strong Doob inequality in the

same way as above by replacing Xn − Xm with X.

With this in mind, we get the following result.

Corollary 3.32 (Burkholder-Gundy inequality). Let 1 < p < ∞ and

f ∈ LpF(Ω;Lrγ,T ). Then we have

E∥∥∥ supt∈[0,T ]

∣∣∣ ∫ t

0

f dβ∣∣∣ ∥∥∥p

rhp,r E

∥∥∥(∫ T

0

|f |2 dt) 1

2∥∥∥pr.

Proof. Since

E∥∥∥ supt∈[0,T ]

∣∣∣ ∫ t

0

f dβ∣∣∣ ∥∥∥p

r≥ E

∥∥∥∫ T

0

f dβ∥∥∥pr

is obvious, the assertion follows from Theorem 3.22 and Theorem 3.31.

3.4 Localization 73

Combining some of the previous results, we get the following representation the-

orem for a special class of martingales.

Theorem 3.33 (Martingale representation theorem). Fix 1 < p <∞,

and let (Mt)t∈[0,T ] be an Lr(U)-valued Lp martingale with respect to the

Brownian filtration Fβ. Then there exists a unique f ∈ LpFβ(Ω;Lrγ,T ) such

that for all t ∈ [0, T ] we have

Mt = M0 +

∫ t

0

f dβ.

Especially, the function t 7→Mt has a continuous version.

Proof. We have MT−M0 ∈ Lp0(Ω,F βT ;Lr(U)). Thus, Theorem 3.27 or Corollary

3.28, respectively, yield a unique f ∈ LpFβ(Ω;Lrγ,T ) such that

MT = M0 +

∫ T

0

f dβ.

Applying Proposition 3.30 leads to

Mt = E[MT |F βt ] = M0 + E

[∫ T

0

f dβ∣∣∣F β

t

]= M0 +

∫ t

0

f dβ.

And the last claim finally follows from Theorem 3.31.

3.4 Localization

So far the stochastic integral has only been defined for integrands that satisfy the

integrability condition E∥∥(∫ T

0|f |2 dt

) 12∥∥pr< ∞. Unfortunately, most functions,

even continuous ones, do not fulfill this property. So, to extend the stochastic

integral in a most natural way, we are going to do a localization argument. As in

the previous sections, we assume that 1 < r <∞ is fixed and that F = (Ft)t∈[0,T ]

is a filtration with the same properties we assumed above.

Definition 3.34. A random variable τ : Ω → I ∪ ∞ with I ∈ N,R+ is

called a stopping time with respect to a filtration (Gi)i∈I if

τ ≤ i ∈ Gi for all i ∈ I.


Proposition 3.35. For every f ∈ LpF(Ω;Lrγ,T ) and every stopping time τ

with values in [0, T ] we have 1[0,τ)f ∈ LpF(Ω;Lrγ,T ) and∫ τ

0

f dβ =

∫ T

0

1[0,τ)f dβ almost surely.

Proof. We first show that 1[0,τ)f is adapted with respect to F = (Ft)t∈[0,T ]. For

this purpose, we first observe that the function

Φf : [0, T ]→ LpF(Ω;Lrγ,T ), Φf (t) = 1[0,t)f,

is continuous by the dominated convergence theorem. For n ∈ N we set

S(n) :=k

2nT : k = 0, . . . , 2n

and τn(ω) := mint ∈ S(n) : t ≥ τ(ω).

Since for each k = 0, . . . , 2n and every k2nT ≤ t < k+1

2nT we have

τn ≤ t = τn ≤ k2nT = τ ≤ k

2nT ∈ F k

2nT⊆ Ft,

we infer that τn is a stopping time with respect to the filtration F. Note that

τn(ω) ≥ τ(ω) and limn→∞ τn(ω) = τ(ω) for all ω ∈ Ω. Moreover, we have for

each n ∈ N and all t ∈ [0, T ]

1τn≤t1[0,τn)f =2n∑k=0

1τn≤t1τn= k2nT1[0, k

2nT )f.

Thus, since f is an adapted process, 1τn≤t1[0,τn)f is Ft-measurable for all t ∈[0, T ]. This implies that

1[0,t∧τn)f = 1τn≤t1[0,τn)f + 1τn>t1[0,t)f

is Ft-measurable. By the continuity of the function Φf and the pointwise con-

vergence of the stopping times, we deduce that

limn→∞

1[0,t∧τn(ω))f(ω) = 1[0,t∧τ(ω))f(ω) = 1[0,t)1[0,τ(ω))f(ω)

for almost all ω ∈ Ω. Hence, 1[0,t)1[0,τ)f is Ft-measurable for all t ∈ [0, T ]. So

far we have shown that 1[0,τ)f is adapted to the filtration F. The first assertion

then follows by

E∥∥∥(∫ T

0

1[0,τ)|f |2 dt) 1

2∥∥∥pr≤ E

∥∥∥(∫ T

0

|f |2 dt) 1

2∥∥∥pr<∞.

3.4 Localization 75

For the integral identity, we first assume that f is an adapted step process of the

form (3.2) with the additional assumption that tj := j2nT for j = 1, . . . , 2n and

n ∈ N fixed. By taking vj :=∑Mj

m=1 x(j)m 1

A(j)m

we thus can write

f =2n∑j=1

1[ j−12n

T, j2nT )

Mj∑m=1

x(j)m 1

A(j)m

=2n∑j=1

1[tj−1,tj)vj.

Now observe that

1[0,τn)f =2n∑j=1

1[0,τn)1[tj−1,tj)vj =2n∑j=1

2n∑k=1

1τn=tk1[0,tk)1[tj−1,tj)vj

=2n∑j=1

2n∑k=j

1τn=tk1[tj−1,tj)vj =2n∑j=1

1[tj−1,tj)1τn≥tjvj.

Hence, since τn ≥ tj = τn ≤ tj−1C ∈ Ftj−1, the process 1[0,τn)f is still an

adapted step process. With this in mind, we obtain∫ T

0

1[0,τn)f dβ =2n∑j=1

1τn≥tjvj(β(tj)− β(tj−1)

)=

2n∑j=1

2n∑k=j

1τn=tkvj(β(tj)− β(tj−1)

)=

2n∑k=1

1τn=tk

k∑j=1

vj(β(tj)− β(tj−1)

)=

2n∑k=1

1τn=tk

∫ tk

0

f dβ

=

∫ τn

0

f dβ.

Now let f ∈ LpF(Ω;Lrγ,T ) and (fn)∞n=1 be an approximating sequence of adapted

step processes for f . As seen in the proof of Lemma 3.21, we can choose these

step processes with dyadic time steps, as assumed above. Then

limn→∞

∥∥1[0,τn)(fn − f)∥∥LpF(Ω;Lrγ,T )

≤ limn→∞

∥∥fn − f∥∥LpF(Ω;Lrγ,T )= 0,

and, using the continuity of Φf , this leads to

limn→∞

∥∥1[0,τn)fn − 1[0,τ)f∥∥LpF(Ω;Lrγ,T )

≤ limn→∞

(‖1[0,τn)fn − 1[0,τn)f‖LpF(Ω;Lrγ,T ) + ‖1[0,τn)f − 1[0,τ)f‖LpF(Ω;Lrγ,T )

)= 0.


Therefore,

limn→∞

∫ T

0

1[0,τn)fn dβ =

∫ T

0

1[0,τ)f dβ in Lp(Ω;Lr(U)).

By Theorem 3.31 we furthermore deduce that

limn→∞

E∥∥∥∫ τn

0

fn dβ −∫ τn

0

f dβ∥∥∥pr≤ lim

n→∞E∥∥∥ supt∈[0,T ]

∣∣∣ ∫ t

0

fn dβ −∫ t

0

f dβ∣∣∣ ∥∥∥p

r

.p,r limn→∞

E∥∥∥∫ T

0

fn dβ −∫ T

0

f dβ∥∥∥pr

= 0.

Thus, we have

limn→∞

E∥∥∥∫ τn

0

fn dβ −∫ τ

0

f dβ∥∥∥pr

≤ limn→∞

(E∥∥∥∫ τn

0

fn dβ −∫ τn

0

f dβ∥∥∥pr

+ E∥∥∥∫ τn

0

f dβ −∫ τ

0

f dβ∥∥∥pr

)= 0,

using the continuity of the stochastic integral process and the pointwise conver-

gence of the stopping times for the second term. We finally conclude that∫ τ

0

f dβ = limn→∞

∫ τn

0

fn dβ = limn→∞

∫ T

0

1[0,τn)fn dβ =

∫ T

0

1[0,τ)f dβ

in Lp(Ω;Lr(U)).

Lemma 3.36. If f ∈ LpF(Ω;Lrγ,T ), then for all δ > 0 and all ε > 0 we have

P(∥∥∥ sup

t∈[0,T ]

∣∣∣ ∫ t

0

f dβ∣∣∣ ∥∥∥

r> ε

)≤ cp,rδ

p

εp+ P

(‖f‖Lrγ,T ≥ δ

)and

P(‖f‖Lrγ,T > ε

)≤ cp,rδ

p

εp+ P

(∥∥∥ supt∈[0,T ]

∣∣∣ ∫ t

0

f dβ∣∣∣ ∥∥∥

r≥ δ

),

where cp,r is the constant from Theorem 3.22.

Proof. By Proposition 3.35 and Theorem 3.22 we have

E∥∥∥ supt∈[0,τ ]

∣∣∣∫ t

0

f dβ∣∣∣ ∥∥∥p

r= E

∥∥∥ supt∈[0,T ]

∣∣∣∫ t∧τ

0

f dβ∣∣∣ ∥∥∥p

r≤ cp,r E‖1[0,τ)f‖pLrγ,T

for all stopping times τ .

3.4 Localization 77

Let δ, ε > 0, and define

µ(ω) := T ∧ inft ∈ [0, T ] :

∥∥∥ sups∈[0,t]

∣∣∣(∫ s

0

f dβ)

(ω)∣∣∣ ∥∥∥

r≥ ε,

ν(ω) := T ∧ inft ∈ [0, T ] : ‖1[0,t)f(ω)‖Lrγ,T ≥ δ

.

Now take τ := µ ∧ ν. Then τ is a stopping time and

E∥∥∥ supt∈[0,T ]

∣∣∣∫ t∧τ

0

f dβ∣∣∣ ∥∥∥p

r≤ εp and E‖1[0,τ)f‖pLrγ,T ≤ δp,

since t 7→ 1[0,t)f and t 7→ sups∈[0,t]

∣∣∫ s0f dβ

∣∣ have continuous paths starting at

zero. Chebyshev’s inequality now leads to

P(∥∥∥ sup

t∈[0,T ]

∣∣∣ ∫ t

0

f dβ∣∣∣ ∥∥∥

r> ε,

∥∥ supt∈[0,T ]

1[0,t)|f |∥∥Lrγ,T

< δ

)≤ P

(∥∥∥ supt∈[0,τ ]

∣∣∣∫ t

0

f dβ∣∣∣ ∥∥∥

r≥ ε

)≤ 1

εpE∥∥∥ supt∈[0,τ ]

∣∣∣∫ t

0

f dβ∣∣∣ ∥∥∥p

r

≤ cp,rεp

E‖1[0,τ)f‖pLrγ,T ≤cp,rδ

p

εp.

Thus, we obtain

P(∥∥∥ sup

t∈[0,T ]

∣∣∣ ∫ t

0

f dβ∣∣∣ ∥∥∥

r> ε

)= P

(∥∥∥ supt∈[0,T ]

∣∣∣ ∫ t

0

f dβ∣∣∣ ∥∥∥

r> ε,

∥∥ supt∈[0,T ]

1[0,t)|f |∥∥Lrγ,T

< δ

)+ P

(∥∥∥ supt∈[0,T ]

∣∣∣ ∫ t

0

f dβ∣∣∣ ∥∥∥

r> ε,

∥∥ supt∈[0,T ]

1[0,t)|f |∥∥Lrγ,T≥ δ

)≤ cp,rδ

p

εp+ P

(∥∥ supt∈[0,T ]

1[0,t)|f |∥∥Lrγ,T≥ δ)

=cp,rδ

p

εp+ P


),

where the last inequality follows from the fact that for each ω ∈ Ω and all

t ∈ [0, T ] we have 1[0,t)|f(ω)| ≤ |f(ω)| with equality for t = T , and therefore,

‖f(ω)‖Lrγ,T = ‖ supt∈[0,T ] 1[0,t)|f(ω)|‖Lrγ,T .

The second inequality follows in the exact same way by interchanging the two

processes. Observe that in this case we still have

E‖1[0,τ)f‖pLrγ,T ≤ cp,r E∥∥∥ supt∈[0,τ ]

∣∣∣∫ t

0

f dβ∣∣∣ ∥∥∥p

r

by Proposition 3.35 and Theorem 3.22.


For a separable Banach space E, we denote by L0(Ω;E) the vector space of all

equivalence classes of measurable functions on Ω with values in the Banach space

E which are identical almost surely. Moreover, we define the map

dP : L0(Ω;E)× L0(Ω;E)→ [0,∞), dP(X, Y ) = E(‖X − Y ‖E ∧ 1).

Proposition 3.37. The pairing (L0(Ω;E), dP) is a complete metric space,

and convergence with respect to this metric coincides with convergence in

probability.

Proof. (1) Let X, Y ∈ L0(Ω;E). Then, of course, dP is well-defined and sym-

metric. Next, if dP(X, Y ) = 0, then ‖X − Y ‖E ∧ 1 = 0 almost surely, which

implies that X = Y almost surely. The converse direction is trivial. Finally,

using that

(a+ b) ∧ 1 ≤ a ∧ 1 + b ∧ 1

for all a, b ≥ 0, we deduce that

dP(X, Y ) ≤ E((‖X − Z‖E + ‖Z − Y ‖E) ∧ 1

)≤ E

((‖X − Z‖E ∧ 1) + (‖Z − Y ‖E ∧ 1)

)= dP(X,Z) + dP(Z, Y )

for all Z ∈ L0(Ω;E). Therefore, (L0(Ω;E), dP) is a metric space.

(2) Next, assume that (Xn)∞n=1 ⊆ L0(Ω;E) with limn→∞ dP(Xn, X) = 0. Then

for every ε ∈ (0, 1) we obtain

limn→∞

P(‖Xn −X‖E > ε

)= lim

n→∞

1

εE1‖Xn−X‖E>ε(ε ∧ 1)

≤ limn→∞

1

εE(‖Xn −X‖E ∧ 1) = lim

n→∞

1

εdP(Xn, X) = 0.

Now, if limn→∞Xn = X in probability, we deduce that for all ε ∈ (0, 1)

limn→∞

dP(Xn, X)

= limn→∞

∫‖Xn−X‖E>ε

‖Xn −X‖E ∧ 1 dP +

∫‖Xn−X‖E≤ε

‖Xn −X‖E ∧ 1 dP

≤ limn→∞

∫‖Xn−X‖E>ε

1 dP +

∫‖Xn−X‖E≤ε

‖Xn −X‖E dP

≤ limn→∞

P(‖Xn −X‖E > ε

)+ ε = ε.

Since ε was arbitrary, we get limn→∞ dP(Xn, X) = 0. And this shows the desired

convergence result.

3.4 Localization 79

(3) At last we show the completeness of the metric space. Let (Xn)∞n=1 ⊆L0(Ω;E) be Cauchy with respect to dP. Then, by the foregoing result, the se-

quence (Xn)∞n=1 is ’Cauchy in probability’, that is

limn,m→∞

P(‖Xn −Xm‖E > ε

)= 0 for all ε > 0.

Therefore, we can find a sequence of uprising integers (kn)∞n=1 with limn→∞ kn =

∞ such that

P(‖Xkn −Xkn+m‖E > 2−n

)< 2−n for all n,m ≥ 1,

which leads to∞∑n=1

P(‖Xkn −Xkn+1‖E > 2−n

)<∞.

By the Borel-Cantelli lemma we get ‖Xkn(ω)−Xkn+1(ω)‖E ≤ 2−n for all ω ∈ Ω0,

where Ω0 ⊆ Ω with P(Ω0) = 1. Hence, for each fixed ω ∈ Ω0, the sequence(Xkn(ω)

)∞n=1

is Cauchy in E and therefore converges to a X(ω) ∈ E. Thus, the

sequence (Xkn)∞n=1 converges almost surely to the random variable X. Then it

does so also in probability and, by the foregoing result, also in the metric dP.

But, since (Xn)∞n=1 is Cauchy with respect to dP, we infer that

limn→∞

dP(Xn, X) ≤ limn→∞

dP(Xn, Xkn) + dP(Xkn , X) = 0,

which means that dP is complete.

As in Definition 3.19, we denote by L0F(Ω;Lrγ,T ) the space of all adapted processes,

i.e., functions f ∈ L0(Ω;Lrγ,T ) with

1[0,t)f ∈ L0(Ω,Ft;Lrγ,t) for all t ∈ [0, T ],

which means that f has a representative f : [0, T ]×Ω×U → R such that 1[0,t)f

is B([0, t))⊗Ft ⊗ Σ-measurable.

Definition 3.38. Let (τn)∞n=1 be a sequence of stopping times with respect

to F and with values in [0, T ], and let f ∈ L0F(Ω;Lrγ,T ). Then we call the

sequence (τn)∞n=1 a localizing sequence for f if

(1) for all ω ∈ Ω there exists an index N(ω) ∈ N such that τn(ω) = T for

all n ≥ N(ω), and

(2) 1[0,τn)f ∈ LpF(Ω;Lrγ,T ) for all n ∈ N.


Remark 3.39. (1) For each f ∈ L0F(Ω;Lrγ,T ) and every stopping time τ with re-

spect to F, the random variable 1[0,τ)f is adapted to F. This was already shown

in the proof of Proposition 3.35.

(2) For every f ∈ L0F(Ω;Lrγ,T ), a localizing sequence (τn)∞n=1 is given by

τn(ω) := infs ∈ [0, T ] : ‖1[0,s)f(ω)‖Lrγ,T ≥ n

,

where we take τn(ω) := T if the infimum is taken over the empty set. Let us

prove this assertion. If we fix an ω ∈ Ω we have

‖1[0,s)f(ω)‖Lrγ,T ≤ ‖f(ω)‖Lrγ,T <∞.

Thus, there exists an index N(ω) such that ‖1[0,s)f(ω)‖Lrγ,T ≤ N(ω) for all 0 ≤s ≤ T . Hence, for each n ≥ N(ω) we have τn(ω) = T . Moreover, we have

‖1[0,τn(ω))f(ω)‖Lrγ,T ≤ n, and therefore

E‖1[0,τn)f‖pLrγ,T ≤ np <∞,

which, combined with the first remark, gives the second property.

For any f ∈ LpF(Ω;Lrγ,T ) we recall from Theorem 3.31 that

t 7→ IβF (1[0,t)f) =

∫ t

0

f dβ

has a continuous version. With this in mind, we can give the following localized

version of Theorem 3.22.

Theorem 3.40 (Ito homeomorphism). The mapping IβF has a unique ex-

tension to a linear mapping

IβF,loc : L0F(Ω;Lrγ,T )→ L0(Ω;Lr(U ;C([0, T ]))),

which is a homeomorphism onto its closed range. Moreover, the estimates

from Lemma 3.36 extend to arbitrary elements f ∈ L0F(Ω;Lrγ,T ).

Proof. Let f ∈ L0(Ω;Lrγ,T ), and let (τn)∞n=1 be the localizing sequence for f from

Remark 3.39 (2). Then,

fn := 1[0,τn)f ∈ LpF(Ω;Lrγ,T ) for all 1 < p <∞,

3.4 Localization 81

and by Theorem 3.31,

IβF (fn) ∈ Lp(Ω;Lr(U ;C([0, T ]))) ⊆ L0(Ω;Lr(U ;C([0, T ])))

for each n ∈ N. Since limn→∞ fn = f almost surely, the sequence (fn)∞n=1 is

’Cauchy in probability’ and, by Lemma 3.36 and Proposition 3.37, we deduce

that(IβF (fn)

)∞n=1

is Cauchy in L0(Ω;Lr(U ;C([0, T ]))). Hence, there exists a limit

X ∈ L0(Ω;Lr(U ;C([0, T ]))). Now define

IβF,loc(f) := X = limn→∞

IβF (fn) in L0(Ω;Lr(U ;C([0, T ]))).

Then IβF,loc is well-defined. Additionally, by Proposition 3.37, we have

limn→∞

IβF (fn) = IβF,loc(f) in probability,

and thus, by passing to a subsequence, we have

limk→∞

IβF (fnk) = IβF,loc(f) almost surely.

Now let ε > 0. Then, by the σ-continuity of the probability measure P, we have

limk→∞

P(∥∥ sup

t∈[0,T ]

|IβF (1[0,t)fnk)|∥∥r> ε)

= P(∥∥ sup

t∈[0,T ]

|IβF,loc(f)(t)|∥∥r> ε),

and similarly, since limk→∞ fnk = f almost surely, we obtain for any δ > 0

limk→∞

P(‖fnk‖Lrγ,T ≥ δ

)= P


).

Next, applying Lemma 3.36 to each fnk , we get

P(∥∥ sup

t∈[0,T ]

|IβF,loc(f)(t)|∥∥r> ε)

= limk→∞

P(∥∥ sup

t∈[0,T ]

|IβF (1[0,t)fnk)|∥∥r> ε)

≤ cp,rδp

εp+ lim

k→∞P(‖fnk‖Lrγ,T ≥ δ

)=cp,rδ

p

εp+ P


).

The other inequality from Lemma 3.36 can be extended in the exact same way.

From this, combined with Proposition 3.37, we infer that IβF,loc is continuous and

has a continuous inverse. This also shows, that the mapping IβF,loc has a closed

range in L0(Ω;Lr(U ;C([0, T ]))).

Now we are prepared to define the stochastic integral for adapted processes.


Definition 3.41. Let f ∈ L0F(Ω;Lrγ,T ) and (fn)∞n=1 be an approximating se-

quence of adapted Lp processes. Then we define the stochastic integral of f

by ∫ ·0

f dβ := IβF,loc(f) = limn→∞

∫ ·0

fn dβ,

where convergence holds in L0(Ω;Lr(U ;C([0, T ]))).

Remark 3.42. (1) The stochastic integral is well-defined in the sense that it is

independent of the approximating sequence.

(2) If f ∈ LpF(Ω;Lrγ,T ) for some 1 < p < ∞, then the integral process which

arises from the Ito integral coincides almost surely with the ’localized’ stochastic

integral process.

(3) This new stochastic integral process is no longer a martingale, but a so called

local martingale (cf. [10] for more information).

Proposition 3.43. For every f ∈ L0F(Ω;Lrγ,T ), each t ∈ [0, T ], and any

stopping time τ with respect to F and with values in [0, T ], we have∫ t∧τ

0

f dβ =

∫ t

0

1[0,τ)f dβ almost surely.

Proof. Let (τn)∞n=1 be the localizing sequence of Remark 3.39 (2), and define

fn := 1[0,τn)f ∈ LpF(Ω;Lrγ,T ).

Then limn→∞ fn = f almost surely and in L0F(Ω;Lrγ,T ). Therefore, by Theorem

3.40, we obtain

limn→∞

E(∥∥∥∫ t∧τ

0

fn dβ −∫ t∧τ

0

f dβ∥∥∥r∧ 1

)≤ lim

n→∞E(∥∥∥ sup

s∈[0,T ]

∣∣∣ ∫ s

0

fn dβ −∫ s

0

f dβ∣∣∣ ∥∥∥

r∧ 1

)= 0.

And since limn→∞ 1[0,τ)fn = 1[0,τ)f in L0F(Ω;Lrγ,T ), we also have for all t ∈ [0, T ]

limn→∞

∫ t

0

1[0,τ)fn dβ =

∫ t

0

1[0,τ)f dβ in L0(Ω;Lr(U)).

3.4 Localization 83

Finally, applying Proposition 3.35 and Corollary 3.25 to each fn, we get∫ t∧τ

0

f dβ = limn→∞

∫ t∧τ

0

fn dβ = limn→∞

∫ T

0

1[0,t∧τ)fn dβ

= limn→∞

∫ T

0

1[0,t)1[0,τ)fn dβ

= limn→∞

∫ t

0

1[0,τ)fn dβ

=

∫ t

0

1[0,τ)f dβ

in L0(Ω;Lr(U)).

Theorem 3.44 (Burkholder-Gundy inequality). Let 1 < p < ∞ and

f ∈ L0F(Ω;Lrγ,T ). Then we have

E∥∥∥ supt∈[0,T ]

∣∣∣ ∫ t

0

f dβ∣∣∣ ∥∥∥p

rhp,r E

∥∥∥(∫ T

0

|f |2 dt) 1

2∥∥∥pr.

This is understood in the sense that the left-hand side is finite if and only if the

right-hand side is finite.

Proof. If the right-hand side is finite, then f ∈ LpF(Ω;Lrγ,T ), and therefore the

inequalities follow from Corollary 3.32. Next, assume the left-hand side is finite.

Let (τn)∞n=1 be the localizing sequence from Remark 3.39 (2), and define

fn := 1[0,τn)f ∈ LpF(Ω;Lrγ,T ).

Then limn→∞ fn = f almost surely. And moreover, by passing to a subsequence,

limk→∞∫ τnk

0f dβ =

∫ T0f dβ almost surely. Thus, the dominated convergence

theorem (here we use the assumption), Proposition 3.43, Theorem 3.22, and

Fatou’s lemma yield

E∥∥∥∫ T

0

f dβ∥∥∥pr

= limk→∞

E∥∥∥∫ τnk

0

f dβ∥∥∥pr

= limk→∞

E∥∥∥∫ T

0

1[0,τnk )f dβ∥∥∥pr

hp,r lim infk→∞

E∥∥∥(∫ T

0

|fnk |2 dt) 1

2∥∥∥pr

≥ E∥∥∥(∫ T

0

|f |2 dt) 1

2∥∥∥pr.

This shows that f ∈ LpF(Ω;Lrγ,T ), and the result again follows from Corollary

3.32.


Finally, we extend Theorem 3.29 to the localized case.

Theorem 3.45. For a measurable and adapted process f : [0, T ] × Ω →Lr(U), which satisfies 〈f, g〉 ∈ L0(Ω;L2([0, T ])) for all g ∈ Lr′(U), the follow-

ing assertions are equivalent:

(1) f ∈ L0F(Ω;Lrγ,T );

(2) there exists a process ζ ∈ L0(Ω;Lr(U ;C([0, T ]))) such that for all g ∈Lr′(U) we have

〈ζ, g〉 =

∫ ·0

〈f, g〉 dβ in L0(Ω;C([0, T ])).

In this situation, we have ζ =∫ ·


Proof. (1) ⇒ (2): By Remark 3.39 and Lemma 3.21, there exists a sequence

(fn)∞n=1 of adapted step processes converging to f in L0F(Ω;Lrγ,T ). Moreover, we

have fn ∈ LpF(Ω;Lrγ,T ) and 〈fn, g〉 ∈ LpF(Ω;L2([0, T ])) for all g ∈ Lr′(U) and some

1 < p < ∞. Therefore, by Theorem 3.29 and Corollary 3.25, we obtain for each

t ∈ [0, T ] ∫ t

0

〈fn, g〉 dβ =⟨∫ t

0

fn dβ, g⟩

almost surely.

Next we define ζ := IβF,loc(f). Then, by definition, we have limn→∞∫ ·

0fn dβ = ζ

in L0(Ω;Lr(U ;C([0, T ]))), and thus

limn→∞

∫ ·0

〈fn, g〉 dβ = 〈ζ, g〉 in L0(Ω;C([0, T ])).

Since U is a σ-finite measure space we can find a sequence (Uk)∞k=1 of disjoint

sets of finite measure such that U =⋃∞k=1 Uk. Similarly as in the proof of The-

orem 3.13, we take U (K) :=⋃Kk=1 Uk, K ∈ N, and we can find an appropriate

subsequence (fnj)∞j=1 such that

limj→∞〈fnj ,1U(K)g〉 = 〈f,1U(K)g〉 in L0

F(Ω;L2([0, T ])).

This leads to∫ ·0

〈f,1U(K)g〉 dβ = limj→∞

∫ ·0

〈fnj ,1U(K)g〉 dβ = 〈ζ,1U(K)g〉

in L0(Ω;C([0, T ])).

3.4 Localization 85

By the dominated convergence theorem we both have limK→∞〈f,1U(K)g〉 = 〈f, g〉in L0

F(Ω;L2([0, T ])) and limK→∞〈ζ,1U(K)g〉 = 〈ζ, g〉 in L0(Ω;C([0, T ])), which

finally yield∫ ·0

〈f, g〉 dβ = limK→∞

∫ ·0

〈f,1U(K)g〉 dβ = limK→∞

〈ζ,1U(K)g〉 = 〈ζ, g〉

in L0(Ω;C([0, T ])).

(2)⇒ (1): We define for each n ∈ N

τn(ω) := T ∧ inft ∈ [0, T ] :

∥∥ sups∈[0,t]

|ζ(ω, s)|∥∥r≥ n

,

which is a stopping time since the map t 7→∥∥sups∈[0,t] |ζ(ω, s)|

∥∥r

is continu-

ous for each ω ∈ Ω by the dominated convergence theorem. Then, ζn :=

ζ(· ∧ τn) ∈ Lp(Ω;Lr(U ;C([0, T ]))) for some 1 < p < ∞ and limn→∞ ζn = ζ

in L0(Ω;Lr(U ;C([0, T ]))). Now fix a g ∈ Lr′(U). By (2) and Proposition 3.43 we

have

〈ζn, g〉 =

∫ ·0

1[0,τn)〈f, g〉 dβ almost surely.

Since ζn ∈ Lp(Ω;Lr(U ;C([0, T ]))), Theorem 3.44 implies that 1[0,τn) 〈f, g〉 ∈Lp(Ω;L2([0, T ])), in particular

〈ζn(T ), g〉 =

∫ T

0

1[0,τn)〈f, g〉 dβ in Lp(Ω).

Since g ∈ Lr′(U) was arbitrary, Theorem 3.29 yield fn := 1[0,τn)f ∈ LpF(Ω;Lrγ,T )

with ζn(T ) =∫ T

0fn dβ. By the same argument applied to ζn(t), combined with

Corollary 3.25, we get

ζn(t) =

∫ t

0

fn dβ for every t ∈ [0, T ].

Thus,(∫ ·

0fn dβ

)∞n=1

converges in L0(Ω;Lr(U ;C([0, T ]))), as we mentioned above.

By Theorem 3.40, the sequence (fn)∞n=1 is Cauchy in L0F(Ω;Lrγ,T ) with limit f ,

and hence, a subsequence converges almost surely to f in Lrγ,T . By the definition

of the stopping times (τn)∞n=1 we also have limn→∞ fn = f almost surely in Lrγ,T .

But this means that f = f ∈ L0F(Ω;Lrγ,T ) and that ζ =

∫ ·0f dβ almost surely.

Chapter 4

An Application to Stochastic

Evolution Equations

In this chapter we apply the theory developed in Chapter 3 to construct new

processes from old ones. On the one hand, we will then be able to establish

a connection to the operator-valued integration theory, and on the other hand,

they are used in the notation of the Ito formula, which will be proved in Section

4.2. In the final section we want to apply Ito’s formula to prove the existence of

solutions for an abstract stochastic evolution equation.

In this chapter we will again assume that 1 < r <∞.

4.1 Ito Processes

Let (βn)∞n=1 be a sequence of independent Brownian motions βn =(βn(t)

)t∈[0,T ]

on a probability space (Ω,A ,P), and let F = (Ft)t∈[0,T ] be a filtration such that

for all 0 ≤ s < t ≤ T and all n ∈ N,

(1) βn(t) is Ft-measurable and

(2) βn(t)− βn(s) is independent of Fs.

Moreover, we assume that x0 : Ω→ Lr(U) is an F0-measurable random variable

and that the functions f : [0, T ] × Ω → Lr(U) and bn : [0, T ] × Ω → Lr(U) are

adapted with respect to F for each n ∈ N satisfying∫ T

0

‖f‖r dt <∞ and∥∥∥(∫ T

0

∞∑n=1

|bn|2 dt) 1

2∥∥∥r<∞ almost surely.

With these notions we have the following definition.

88 An Application to Stochastic Evolution Equations

Definition 4.1. Under the assumptions taken above, we call an Lr(U)-

valued process X =(X(t)

)t∈[0,T ]

an Ito process (or standard process) if it

has the integral representation

X(t) = x0 +

∫ t

0

f ds+∞∑n=1

∫ t

0

bn dβn for 0 ≤ t ≤ T. (4.1)

Remark 4.2. (1) Instead of (4.1), we also formally write

dX = f dt+∞∑n=1

bn dβn.

(2) The function f is almost surely Bochner-integrable, and therefore the paths

of f belong to L1([0, T ], Lr(U)). Moreover∥∥∥∫ T

0

|f | dt∥∥∥r≤∫ T

0

‖f‖r dt <∞,

which means that f also belongs to L0F(Ω;Lr(U ;L1([0, T ]))).

(3) By the assumptions we made on the filtration, each stochastic integral in

(4.1) is well-defined.

Next, we give an answer to the question why the series in (4.1) is well-defined.

To prove this, we first need some preliminary results where we always assume

that x0 = 0 almost surely.

Lemma 4.3. In addition to the above properties, let bn ∈ LpF(Ω;Lrγ,T ) for

each n ∈ N. Then we have for any fixed N ∈ N

E∥∥∥ N∑n=1

∫ T

0

bn dβn

∥∥∥prhp,r E

∥∥∥(∫ T

0

N∑n=1

|bn|2 dt) 1

2∥∥∥pr.

Proof. First consider adapted step processes

bn =M∑m=1

v(n)m 1[tm−1,tm),

where 0 = t0 < . . . < tM = T and v(n)m : Ω→ Lr(U) is a simple function, which is

Ftm−1-measurable for each m = 1, . . . ,M and n = 1, . . . , N .

4.1 Ito Processes 89

Now we define for m = 1, . . . ,M and n = 1, . . . , N

ξ(m−1)N+n := βn(tm)− βn(tm−1), F(m−1)N+n := σ(ξ` : ` ≤ (m− 1)N + n),

γ(m−1)N+n := (tm − tm−1)−12 ξ(m−1)N+n, v(m−1)N+n := v(n)

m .

Then, for all k = 1, . . . ,M ·N , ξk is a centered, Fk-measurable random variable,

which is independent of Fk−1. Additionally, vk is Fk−1-measurable, which means

that the sequence (vk)MNk=1 is predictable with respect to the filtration (Fk)

MNk=1 .

And furthermore, by the independence of the Brownian motions, (γk)MNk=1 is a

sequence of independent standard Gaussian variables.

By applying the Decoupling theorem and the Kahane inequality as in Corollary

2.24, we obtain

E∥∥∥ N∑n=1

∫ T

0

bn dβn

∥∥∥pr

= E∥∥∥ N∑n=1

M∑m=1

v(n)m

(βn(tm)− βn(tm−1)

) ∥∥∥pr

= E∥∥∥MN∑k=1

vkξk

∥∥∥pr

hp,r E∥∥∥(MN∑

k=1

|vk|2(td kNe − td k

Ne−1)

) 12∥∥∥pr

= E∥∥∥( N∑

n=1

M∑m=1

|v(n)m |2(tm − tm−1)

) 12∥∥∥pr

= E∥∥∥(∫ T

0

N∑n=1

|bn|2 dt) 1

2∥∥∥pr.

The general estimate finally follows by approximating each adapted Lp process

bn with adapted step processes.

Proposition 4.4. Let (bn)∞n=1 ⊆ L0F(Ω;Lrγ,T ) such that

E∥∥∥(∫ T

0

∞∑n=1

|bn|2 dt) 1

2∥∥∥pr<∞

for some 1 < p < ∞. Then for each n ∈ N, the integral Xn(t) :=∫ t

0bn dβn

exists in Lp(Ω,Ft;Lr(U)), and the sum X(t) :=

∑∞n=1 Xn(t) converges in

Lp(Ω,Ft;Lr(U)) for each fixed t ∈ [0, T ]. Moreover, we have

E‖X(t)‖pr hp,r E∥∥∥(∫ t

0

∞∑n=1

|bn|2 dt) 1

2∥∥∥pr.


Proof. By the assumption we infer that bn ∈ LpF(Ω;Lrγ,T ), and hence Xn(t) is an

element of Lp(Ω,Ft;Lr(U)) for all n ∈ N and each t ∈ [0, T ]. Furthermore, by

Lemma 4.3, we obtain

limM,N→∞

E∥∥∥ N∑n=M

Xn(t)∥∥∥prhp,r lim

M,N→∞E∥∥∥(∫ t

0

N∑n=M

|bn|2 dt) 1

2∥∥∥pr

= 0.

So,(∑N

n=1Xn(t))∞N=1

is Cauchy in Lp(Ω,Ft;Lr(U)), which gives the desired con-

vergence result. Finally, by Lemma 4.3 and the dominated convergence theorem,

we obtain

E‖X(t)‖pr = limN→∞

E∥∥∥ N∑n=1

Xn(t)∥∥∥prhp,r lim

N→∞E∥∥∥(∫ t

0

N∑n=1

|bn|2 dt) 1

2∥∥∥pr

= E∥∥∥(∫ t

0

∞∑n=1

|bn|2 dt) 1

2∥∥∥pr.

Proposition 4.5. Under the assumptions of Proposition 4.4, the process(X(t)

)t∈[0,T ]

is an Lp martingale with respect to F and has a continuous

version satisfying the maximal inequality

E∥∥ supt∈[0,T ]

|X(t)|∥∥pr.p,r E‖X(T )‖pr.

Proof. By Proposition 4.4, X(t) is Ft-measurable, and by Proposition 3.30,

each process(Xn(t)

)t∈[0,T ]

is an Lp martingale with respect to F. Hence, by the

continuity of the conditional expectation, we get

E[X(t)|Fs] = limN→∞

E[ N∑n=1

Xn(t)|Fs

]= lim

N→∞

N∑n=1

E[Xn(t)|Fs]

= limN→∞

N∑n=1

Xn(s) = X(s)

in Lp(Ω,Fs;Lr(U)) for each fixed 0 ≤ s ≤ t ≤ T . Thus,

(X(t)

)t∈[0,T ]

is an

Lp martingale. By Theorem 3.31, the process YN :=∑N

n=1Xn has a continuous

version and, by proceeding in the exact same way as in the proof of that theorem,

we obtain a continuous version of(X(t)

)t∈[0,T ]

satisfying the desired maximal

inequality.

We next consider the localized case.


Lemma 4.6. Under the assumptions of Proposition 4.4, we have for all δ > 0

and all ε > 0

P(∥∥ sup

t∈[0,T ]

|X(t)|∥∥r> ε)≤ cp,rδ

p

εp+ P

(∥∥∥( ∞∑n=1

|bn|2) 1

2∥∥∥Lrγ,T

≥ δ

)and

P(∥∥∥( ∞∑

n=1

|bn|2) 1

2∥∥∥Lrγ,T

> ε

)≤ cp,rδ

p

εp+ P

(∥∥ supt∈[0,T ]

|X(t)|∥∥r≥ δ),

where cp,r is the constant from Proposition 4.4.

Proof. Since for every stopping time τ we have

E∥∥∥(∫ T

0

1[0,τ)

∞∑n=1

|bn|2 dt) 1

2∥∥∥pr≤ E

∥∥∥(∫ T

0

∞∑n=1

|bn|2 dt) 1

2∥∥∥pr<∞,

we obtain by Proposition 4.4

E‖X(τ)‖pr = E∥∥∥ ∞∑n=1

∫ τ

0

bn dβn

∥∥∥pr

= E∥∥∥ ∞∑n=1

∫ T

0

1[0,τ)bn dβn

∥∥∥pr

hp,r E∥∥∥(∫ T

0

1[0,τ)

∞∑n=1

|bn|2 dt) 1

2∥∥∥pr

= E∥∥1[0,τ)‖b‖l2

∥∥pLrγ,T

.

Now, the proof can be finished in the same way as in Lemma 3.36.

Theorem 4.7. Let X be an Ito process given by dX =∑∞

n=1 bn dβn. Then,

the process X is well-defined as an element of L0(Ω;Lr(U ;C([0, T ]))). More-

over, the assertions from Lemma 4.6 extend to this process.

Proof. For any fixed N ∈ N and each n = 1, . . . , N , let (τ(n)k )∞k=1 be the localizing

sequence for bn ∈ L0F(Ω;Lrγ,T ) from Remark 3.39 (2). Then b

(n)k := 1

[0,τ(n)k )

bn ∈

LpF(Ω;Lrγ,T ) for some 1 < p < ∞, and limk→∞ b(n)k = bn in L0

F(Ω;Lrγ,T ) for all

n = 1, . . . , N . By definition we have

limk→∞

∫ ·0

b(n)k dβn =

∫ ·0

bn dβn in L0(Ω;Lr(U ;C([0, T ]))),


and therefore, by the definition of this metric space,

limk→∞

N∑n=1

∫ ·0

b(n)k dβn =

N∑n=1

∫ ·0

bn dβn in L0(Ω;Lr(U ;C([0, T ]))).

Let ε > 0 and δ > 0. By passing to a subsequence, we obtain similarly as in the

proof of Theorem 3.40

limj→∞

P(∥∥∥ sup

t∈[0,T ]

∣∣∣ N∑n=1

∫ t

0

b(n)kj

dβn

∣∣∣ ∥∥∥r> ε

)= P

(∥∥∥ supt∈[0,T ]

∣∣∣ N∑n=1

∫ t

0

bn dβn

∣∣∣ ∥∥∥r> ε

)and

limj→∞

P(∥∥∥( N∑

n=1

|b(n)kj|2) 1

2∥∥∥Lrγ,T

≥ δ

)= P

(∥∥∥( N∑n=1

|bn|2) 1

2∥∥∥Lrγ,T

≥ δ

).

Using this together with Lemma 4.6 for each sum∑N

n=1 b(n)kj

, we get

P(∥∥∥ sup

t∈[0,T ]

∣∣∣ N∑n=1

∫ t

0

bn dβn

∣∣∣ ∥∥∥r> ε

)≤ cp,rδ

p

εp+ P

(∥∥∥( N∑n=1

|bn|2) 1

2∥∥∥Lrγ,T

≥ δ

)and similarly

P(∥∥∥( N∑

n=1

|bn|2) 1

2∥∥∥Lrγ,T

> ε

)≤ cp,rδ

p

εp+ P

(∥∥∥ supt∈[0,T ]

∣∣∣ N∑n=1

∫ t

0

bn dβn

∣∣∣ ∥∥∥r≥ δ

).

Since ∥∥∥(∫ T

0

∞∑n=1

|bn|2 dt) 1

2∥∥∥r<∞ almost surely,

we have

limN,M→∞

P(∥∥∥( N∑

n=M

|bn|2) 1

2∥∥∥Lrγ,T

≥ δ

)= 0.

Thus, by the previous estimate,(∑N

n=1

∫ ·0bn dβn

)∞N=1

is ’Cauchy in probability’,

and so also in L0(Ω;Lr(U ;C([0, T ]))). Hence, by Proposition 3.37, the series∑∞n=1

∫ ·0bn dβn is convergent in this space, i.e., the Ito process

(X(t)

)t∈[0,T ]

is well-

defined. Finally, a limiting argument also shows that the assertions of Lemma

4.6 are still true for this kind of process.

Finally, we extend Theorem 3.45 to Ito processes.


Theorem 4.8. For each n ∈ N, let bn : [0, T ] × Ω → Lr(U) be measurable

and adapted to a filtration F, and let(〈bn, g〉

)∞n=1∈ L0(Ω;L2([0, T ]; l2)) for

all g ∈ Lr′(U). Then the following assertions are equivalent:

(1)∥∥(∫ T

0

∑∞n=1 |bn|2 dt

) 12∥∥r<∞ almost surely;

(2) there exists a process X ∈ L0(Ω;Lr(U ;C([0, T ]))) such that for all

g ∈ Lr′(U) we have

〈X, g〉 =∞∑n=1

∫ ·0

〈bn, g〉 dβn in L0(Ω;C([0, T ])).

In this situation, X equals almost surely the Ito process formally given by

dX =∑∞

n=1 bn dβn.

Proof. (1)⇒ (2) : The assumption implies that bn ∈ L0F(Ω;Lrγ,T ) for each n ∈ N.

Thus, from Theorem 3.45 we deduce that for any n ∈ N⟨∫ ·0

bn dβn, g⟩

=

∫ ·0

〈bn, g〉 dβn in L0(Ω;C([0, T ])).

Finally, by taking X :=∑∞

n=1

∫ ·0bn dβn ∈ L0(Ω;Lr(U ;C([0, T ]))), we have

〈X, g〉 =∞∑n=1

⟨∫ ·0

bn dβn, g⟩

=∞∑n=1

∫ ·0

〈bn, g〉 dβn in L0(Ω;C([0, T ])),

where we used that∑∞

n=1

∫ ·0bn dβn also converges in L0(Ω;C([0, T ];Lr(U))).

(2) ⇒ (1) : For m ∈ N let Pm : Lr(U) → Lr(U) be the operator from Lemma

A.26 (see also the proof of Theorem 3.29). Then, for any k = 1, . . . , Nm, we have

1U

(m)k∈ Lr′(U) and

⟨Pmbn(t, ω),1

U(m)k

⟩=

∫U

(m)k

bn(t, ω) dµ =⟨bn(t, ω),1

U(m)k

⟩for all (t, ω) ∈ [0, T ]×Ω and each n ∈ N. By the assumptions, almost surely this

yields ∥∥∥(∫ T

0

∞∑n=1

∣∣∣ ∫U

(m)k

bn dµ1U

(m)k

∣∣∣2 dt) 1

2∥∥∥r

= µ(U(m)k )

1r

(∫ T

0

∞∑n=1

∣∣⟨bn,1U(m)k

⟩∣∣2 dt) 1

2<∞.


Therefore, by the definition of Pm, we obtain∥∥(∫ T

0

∑∞n=1 |Pmbn|2 dt

) 12∥∥r< ∞,

and by Theorem 4.7 we deduce that the series∑∞

n=1

∫ ·0Pmbn dβn converges in

L0(Ω;Lr(U ;C([0, T ]))). Thus, we obtain

⟨ ∞∑n=1

∫ ·0

Pmbn dβn, g⟩

=∞∑n=1

⟨∫ ·0

Pmbn dβn, g⟩

almost surely.

Now let g ∈ Lr′(U) be arbitrary, and observe that also Pmg ∈ Lr′(U). Then, by

(2) and a small computation similar as in the proof of Theorem 3.29, we get

〈PmX, g〉 = 〈X,Pmg〉 =∞∑n=1

∫ ·0

〈bn, Pmg〉 dβn =∞∑n=1

∫ ·0

〈Pmbn, g〉 dβn

=∞∑n=1

⟨∫ ·0

Pmbn dβn, g⟩

=⟨ ∞∑n=1

∫ ·0

Pmbn dβn, g⟩

almost surely. By Corollary A.8, this gives PmX =∑∞

n=1

∫ ·0Pmbn dβn almost

surely. Moreover, by the properties of the operators Pm and the dominated con-

vergence theorem, we have limm→∞ PmX = X in L0(Ω;Lr(U ;C([0, T ]))). By

taking cm := (Pmbn)∞n=1 ∈ L0(Ω;Lr(U ;L2([0, T ]; l2))), Theorem 4.7 yield that

(cm)∞m=1 is Cauchy in L0(Ω;Lr(U ;L2([0, T ]; l2))) and therefore has a limit c. Sim-

ilar as in Theorem 3.29 and Theorem 3.45, we can show that

limm→∞

cm,n(t, ω, u) = limm→∞

(Pmbn(t, ω))(u) = bn(t, ω, u) and

limm→∞

cm,n(t, ω, u) = cn(t, ω, u)

almost everywhere and for any n ∈ N. Finally, this leads to (bn)∞n=1 = c ∈L0(Ω;Lr(U ;L2([0, T ]; l2))) and this yield X =

∑∞n=1

∫ ·0bn dβn almost surely.

Remark 4.9. Let D′ be a dense linear subspace of Lr′(U). Then the previous

theorem remains valid if we replace Lr′(U) with D′. To prove this, fix a g ∈ Lr′(U)

and choose (gm)∞m=1 ⊆ D′ such that g = limm→∞ gm. Then of course we have

〈X, g〉 = limm→∞〈X, gm〉 in L0(Ω;C([0, T ])). Since (2) holds for every gm, this

convergence and Theorem 4.7 imply that (am)∞m=1 ⊆ L0(Ω;L2([0, T ]; l2)), given by

am :=(〈bn, gm〉

)∞n=1

, is Cauchy in L0(Ω;L2([0, T ]; l2)). Similar as in the previous

proof, we can show that the obtaining limit equals(〈bn, g〉

)∞n=1

almost surely,

and thus(〈bn, g〉

)∞n=1∈ L0(Ω;L2([0, T ]; l2)). Finally, by this convergence and

Theorem 4.7, we have

∞∑n=1

∫ ·0

〈bm, g〉 dβn = limm→∞

∞∑n=1

∫ ·0

〈bn, gm〉 dβn = limm→∞

〈X, gm〉 = 〈X, g〉

in L0(Ω;C([0, T ]), and Theorem 4.8 concludes the proof.


We next make a short excursion in the theory of stochastic integration of operator-

valued processes with respect to an H-cylindrical Brownian motion. Here, we

assume that(H, (·|·)H

)is a real separable Hilbert space.

Definition 4.10. We call a family WH = (WH(t))t∈[0,T ] of bounded linear

operators from H to L2(Ω) an H-cylindrical Brownian motion if:

(1) WHh = (WH(t)h)t∈[0,T ] is a real-valued Brownian motion for each h ∈ Hwith ‖h‖H = 1;

(2) E(WH(s)g ·WH(t)h

)= (s ∧ t)(g|h)H for all s, t ∈ [0, T ] and g, h ∈ H.

The procedure is similar to that in this thesis. We first consider finite rank

adapted step processes Φ: [0, T ] × Ω → B(H,Lr(U)) with respect to a given

filtration F = (Ft)t∈[0,T ], i.e.,

Φ(t, ω) =M∑m=1

N∑n=1

1[tn−1,tn)(t)1Amn(ω)k∑j=1

hj ⊗ xjmn,

where 0 = t0 < . . . < tN = T , for each n = 1, . . . , N the sets A1n, . . . , AMn

are disjoint and belong to Ftn−1 , the vectors h1, . . . , hk ∈ H are orthonormal,

and the vectors xjmn belong to Lr(U). For h ∈ H and x ∈ Lr(U) we denote by

h⊗ x ∈ B(H,Lr(U)) the operator defined by

(h⊗ x)h′ := (h|h′)H x, h′ ∈ H.

Moreover, we assume that WH is an H-cylindrical Brownian motion on (Ω,A ,P),

with the properties that for all h ∈ H

(1) WH(t)h is Ft-measurable and

(2) WH(t)h−WH(s)h is independent of Fs for t > s.

A filtration satisfying these conditions is given by FWH = (FWHt )t∈[0,T ], where

FWHt := σ

(WH(s)en : s ≤ t, n ∈ N

)and (en)∞n=1 is an orthonormal basis of H.

The stochastic integral with respect to WH of a finite rank adapted step process

Φ of the above form is then defined as∫ T

0

Φ dWH :=M∑m=1

N∑n=1

1Amn

k∑j=1

(WH(tn)hj −WH(tn−1)hj

)xjmn.

By approximation, the stochastic integral can now be extended to operator-valued

adapted processes. We will not go into any further detail here and refer to [10]

and [15] for more information on this theory.


Interestingly, there is a connection between operator-valued stochastic integrals

and Ito processes.

Theorem 4.11 (Operator-valued integrals). LetWH be anH-cylindrical

Brownian motion, and let B : [0, T ] × Ω → B(H,Lr(U)) be an operator-

valued process, which is adapted to the filtration F and satisfies B∗g ∈L0(Ω;L2([0, T ];H)) for each g ∈ Lr′(U). Moreover, we assume that

‖B‖γ([0,T ];H,Lr(U)) hp,r

∥∥∥(∫ T

0

∞∑n=1

|Ben|2 dt) 1

2∥∥∥r<∞ almost surely,

where (en)∞n=1 is an orthonormal basis of H (cf. [15] for the norm-equivalence

and the definition of the norm on the left-hand side). Then the process

X(t) :=

∫ t

0

B dWH

equals the Lr(U)-valued Ito process given by

dX =∞∑n=1

bn dβn,

where the sequences b = (bn)∞n=1 and β = (βn)∞n=1 are defined by bn := Benand βn := WHen.

Proof. By [15, Corollary 4.5.10], the operator-valued stochastic integral has the

following series representation

X(t) =

∫ t

0

B dWH =∞∑n=1

∫ t

0

Ben d(WHen)

with convergence in L0(Ω;Lr(U)) for each t ∈ [0, T ]. Thus, we obtain

X(t) =∞∑n=1

∫ t

0

Ben d(WHen) =∞∑n=1

∫ t

0

bn dβn.

By the definition of WH , (βn)∞n=1 = (WHen)∞n=1 is a sequence of independent

Brownian motions, and since B is adapted to F, Lemma 3.5.1 in [15] implies that

each bn is adapted with respect to F. Finally, since∥∥(∫ T

0

∑∞n=1 |bn|2 dt

) 12∥∥r<∞

almost surely, X is an Lr(U)-valued Ito process, and the series even converges in

L0(Ω;Lr(U ;C([0, T ]))), by Theorem 4.7.

And a converse direction is also true.


Theorem 4.12. Let X be an Ito Process given by dX =∑∞

n=1 bn dβn,

and assume that(〈bn, g〉

)∞n=1∈ L0(Ω;L2([0, T ]; l2)) for all g ∈ Lr

′(U) and∥∥(∑∞

n=1 |bn(ω, t)|2) 1

2∥∥r< ∞ for almost all (ω, t) ∈ Ω × [0, T ]. Then X can

be represented as an operator-valued integral

X =

∫ ·0

B dWH ,

where H = l2, and the H-cylindrical Brownian motion and the operator-

valued function B : [0, T ]× Ω→ B(H,Lr(U)) are given by

WH

((hn)∞n=1

)=∞∑n=1

hnβn,

B(t, ω)((hn)∞n=1

)=∞∑n=1

hnbn(t, ω).

Proof. We only have to show that WH is indeed an H-cylindrical Brownian

motion, and that B is well-defined with ‖B‖γ([0,T ];H,Lr(U)) <∞. For this purpose,

let g, h ∈ l2 and 0 ≤ s ≤ t ≤ T be arbitrary. Observe that βn(t) is independent

of βm(s) for all m 6= n, and that βn(t)− βn(s) is a (centered) Gaussian variable

with variance t− s for any n ∈ N. Thus, we obtain

E(WH(s)g ·WH(t)h

)=∞∑n=1

Egnhnβn(s)βn(t) +∑n6=m

Egmhnβm(s)βn(t)

=∞∑n=1

gnhhEβn(s)βn(t) +∑n6=m

gmhnEβm(s)Eβn(t)

= s(g|h)l2 ,

using that

Eβn(s)βn(t) = 12

(Eβn(t)2 + Eβn(s)2 − E

(βn(t)− βn(s)

)2)

= 12(t+ s− t+ s) = s.

This implies E|WH(t)h|2 = t‖h‖l2 < ∞. And since linearity is trivial, this

means that WH(t) is well-defined as a bounded operator from H to L2(Ω) with

‖WH(t)‖B(H,L2(Ω)) =√t. Next, we show that (WH(t)h)t∈[0,T ] is a Brownian mo-

tion if we further assume that ‖h‖l2 = 1. Clearly, we have

WH(0)h =∞∑n=1

hnβn(0) = 0 almost surely.


For N ∈ N and 0 ≤ s ≤ t ≤ T we define

X t,sN :=

N∑n=1

hn(βn(t)− βn(s)

).

Since (βn)∞n=1 is a sequence of independent Brownian motions, X t,sN is a Gaus-

sian random variable with variance qN := (t − s)∑N

n=1 h2n. Moreover, we have

limN→∞Xt,sN = WH(t)h −WH(s)h in L2(Ω), and by Proposition A.17 we infer

that WH(t)h−WH(s)h is Gaussian with variance

q = limN→∞

qN = (t− s)‖h‖2l2 = t− s.

Additionally, by the independence of the Brownian motions, WH(t)h −WH(s)h

is independent of WH(r)h whenever 0 ≤ r ≤ s. So, WH is an H-cylindrical

Brownian motion.

Next, we inspect the operator B. Let h = (hn)∞n=1 ∈ l2. By assumption, we have

for all (t, ω) ∈ [0, T ]× Ω

‖B(t, ω)h‖r =∥∥∥ ∞∑n=1

hnbn(t, ω)∥∥∥r≤ ‖h‖l2

∥∥∥( ∞∑n=1

|bn(ω, t)|2) 1

2∥∥∥r<∞,

which implies that B(t, ω) is a bounded operator for all (t, ω) ∈ [0, T ] × Ω.

Moreover, by Fubini’s theorem, we obtain

〈Bh, g〉 =

∫U

∞∑n=1

hnbng dµ =∞∑n=1

hn

∫U

bng dµ = (h|B∗g)l2

for all h ∈ l2 and g ∈ Lr′(U). Thus, the adjoint operator B∗ : Lr

′(U) → l2 is

given by B∗g =(〈bn, g〉

)∞n=1

, and almost surely we have

‖B∗g‖2L2([0,T ],l2) =

∫ T

0

∞∑n=1

〈bn, g〉2 dt <∞.

Finally, the operator B satisfies

‖B‖γ([0,T ];H,Lr(U)) hp,r

∥∥∥(∫ T

0

∞∑n=1

|bn|2 dt) 1

2∥∥∥r<∞

such that Theorem 4.11 concludes the proof.

Remark 4.13. As a conclusion we observe that an Lr(U)-valued Ito process,

given by dX =∑∞

n=1 bn dβn, is nothing but an operator-valued integral expanded

with respect to an orthonormal basis of a Hilbert space H.

4.2 The Ito Formula 99

4.2 The Ito Formula

By computing a deterministic integral we nearly always use the fundamental the-

orem of calculus since only a few integrals can be done comfortably by applying

the definition of the integral. In the case of a stochastic integral it is very similar,

and thus it would be quite neat if we would have an analogue to the fundamental

theorem of calculus. In the case where Lr(U) = RN for an N ∈ N, there exists

the well-known Ito formula, which can be found in many books about stochastic

integration (cf. [1, (5.3.8)] for the proof of the following proposition).

From this point on, we return to general Ito processes as defined in (4.1).

Proposition 4.14 (Ito’s formula - the finite-dimensional case).

Fix M,N ∈ N. Let Φ: [0, T ]× RN → R be an element of C1,2([0, T ]× RN),

(βm)Mm=1 be an M -dimensional Brownian motion, and let X be an RN -valued

Ito process given by dX = f dt +∑M

m=1 bm dβm. Then, almost surely we

have for all t ∈ [0, T ]

Φ(t,X(t)

)= Φ

(0, X(0)

)+

∫ t

0

∂tΦ(s,X(s)

)ds+

∫ t

0

D2Φ(s,X(s)

)f(s) ds

+M∑m=1

∫ t

0

D2Φ(s,X(s)

)bm(s) dβm(s)

+1

2

∫ t

0

M∑m=1

(D2

2Φ(s,X(s)

)bm(s)

)bm(s) ds.

To extend this formula to functions Φ: [0, T ] × Lr(U1) → Ls(U2) and Lr(U1)-

valued Ito processes, we need the following lemma.

Lemma 4.15. Let X be an Ito process given by dX = f dt+∑M

m=1 bm dβm,

and additionally let bm ∈ L0(Ω;L2([0, T ];Lr(U))) for each m = 1, . . . ,M .

Then there exist sequences (fn)∞n=1 and (b(m)n )∞n=1 of adapted step processes

such that

f = limn→∞

fn in L0(Ω;L1([0, T ];Lr(U))) and

bm = limn→∞

b(m)n in L0(Ω;L2([0, T ], Lr(U))) ∩ L0(Ω;Lrγ,T )

for all m = 1, . . . ,M .


Proof. (1) Similar as in the proof of Lemma 3.21, we define for n ∈ N

fn(t, ω) :=(Anfδn(ω)

)(t) =

2n∑j=1

1[(j−1)T

2n, jT2n

)(t)

2n

T

∫ jT2n

(j−1)T2n

f(s− δn, ω) ds,

where An is the operator from Proposition 3.12, fδn := f(· − δn), and δn := T2n

.

As in the proof of that Lemma, we deduce that

vj,n :=

∫ jT2n

(j−1)T2n

f(s− δn, ·) ds

is an F (j−1)T2n

-measurable Lr(U)-valued random variable, i.e., fn is adapted to the

filtration F. Observe that f, fδn ∈ L0(Ω;L1([0, T ];Lr(U))), and by the dominated

convergence theorem, we have

limn→∞

‖fδn − f‖L1([0,T ];Lr(U)) = 0.

Therefore, by Proposition 3.12, almost surely we have

limn→∞

‖fn − f‖L1([0,T ];Lr(U))

≤ limn→∞

(‖fn − Anf‖L1([0,T ];Lr(U)) + ‖Anf − f‖L1([0,T ];Lr(U))

)≤ lim

n→∞

(‖fδn − f‖L1([0,T ];Lr(U)) + ‖Anf − f‖L1([0,T ];Lr(U))

)= 0,

and hence the convergence also holds in L0(Ω;L1([0, T ];Lr(U))). In general, the

processes fn are not adapted step processes. Thus, by approximating each vj,n in

probability by a sequence of F (j−1)T2n

-measurable simple functions, as in the proof

of Lemma 3.21, we obtain a sequence of adapted step processes (fn,m)∞m=1 such

that limm→∞ fn,m = fn in L0(Ω;L1([0, T ];Lr(U))). By taking an appropriate sub-

sequence (mn)∞n=1, we finally obtain the required sequence (fn)∞n=1 = (fn,mn)∞n=1.

(2) Fix an m ∈ 1, . . . ,M. We then define the sequence (b(m)n )∞n=1 as in the first

part of this proof (just replace f by bm). Since bm ∈ L0(Ω;L2([0, T ];Lr(U))),

Proposition 3.12 implies that

limn→∞

b(m)n = bm in L0(Ω;L2([0, T ];Lr(U))).

Since bm ∈ L0(Ω;Lrγ,T ), we next can apply an embedding argument similar as in

the proof of Lemma 3.21 to obtain

limn→∞

b(m)n = bm in L0(Ω;Lrγ,T ).


In the next theorem we assume that 1 < s < ∞, and that (U1,Σ1, µ1) and

(U2,Σ2, µ2) are σ-finite measure spaces with countably generated σ-algebras Σ1

and Σ2.

Theorem 4.16 (Ito’s formula). Assume that Φ: [0, T ]×Lr(U1)→ Ls(U2)

is an element of C1,2([0, T ] × Lr(U1);Ls(U2)), (βm)∞m=1 is a sequence of in-

dependent Brownian motions, and X is an Lr(U1)-valued Ito process given by

dX = f dt+∑∞

m=1 bm dβm. Further, let (bm)∞m=1 ∈ L0(Ω;L2([0, T ]; l2(Lr(U1)))).

Then, almost surely we have for all t ∈ [0, T ]

Φ(t,X(t)

)= Φ

(0, X(0)

)+

∫ t

0

∂tΦ(s,X(s)

)ds+

∫ t

0

D2Φ(s,X(s)

)f(s) ds

+∞∑m=1

∫ t

0

D2Φ(s,X(s)

)bm(s) dβm(s) (4.2)

+1

2

∫ t

0

∞∑m=1

(D2

2Φ(s,X(s)

)bm(s)

)bm(s) ds.

Remark 4.17. In the situation of Theorem 4.16, the definition of an Ito process

implies that (bm)∞m=1 ∈ L0(Ω;Lr(U1;L2([0, T ]; l2))). If we furthermore assume

that r ≤ 2, Minkowski’s integral inequality yield

Lr(U1;L2([0, T ]; l2)) ⊆ L2([0, T ]; l2(Lr(U1))).

Therefore, the assumption that (bm)∞m=1 ∈ L0(Ω;L2([0, T ]; l2(Lr(U1)))) is auto-

matically fulfilled. On the other hand, if r ≥ 2, then we have

L2([0, T ]; l2(Lr(U1))) ⊆ Lr(U1;L2([0, T ]; l2))

and the additional assumption will be necessary.

Proof (of Theorem 4.16). The proof is divided into four steps.

(1) We first show that (4.2) is true if Φ is an R-valued function and we have an

M -dimensional Brownian motion as well as an Ito process X represented by a

simple function x0 and adapted step processes f and (bm)Mm=1. In this case, the

integer

N := max

dim(R(x0)

), dim

(R(f)

), dim

(R(b1)

), . . . , dim

(R(bM)

)is finite, which means that these processes take their values in an N -dimensional

subspace E of Lr(U1). Therefore, we have an isomorphism I : E → RN .


Since f and each bm are adapted step processes, almost surely we trivially have∫ T

0

‖I(f)‖RN dt <∞ and∥∥∥(∫ T

0

M∑m=1

|I(bm)|2 dt) 1

2∥∥∥RN

<∞

Additionally, by a direct computation using the linearity of I, almost surely we

obtain

I(X) = I(x0) +

∫ ·0

I(f(s)) ds+M∑m=1

∫ ·0

I(bm(s)) dβm(s).

So, I(X) is a well-defined Ito process. We next define the function

Φ : [0, T ]× RN → R, Φ(t, x) = Φ(t, I−1(x)).

Then Φ ∈ C1,2([0, T ]× RN) with

∂tΦ(t, x) = ∂tΦ(t, I−1(x)

),

D2Φ(t, x) : RN → R, h 7→ D2Φ(t, I−1(x)

)I−1(h),

D22Φ(t, x) : RN → B(RN ,R) h 7→

(D2

2Φ(t, I−1(x)

)I−1(h)

)I−1.

Finally, Proposition 4.14 leads to

Φ(t,X(t)

)= Φ

(t, I(X(t))

)= Φ

(0, I(X(0))

)+

∫ t

0

∂tΦ(s, I(X(s))

)ds

+

∫ t

0

D2Φ(s, I(X(s))

)I(f(s)) ds

+∞∑m=1

∫ t

0

D2Φ(s, I(X(s))

)I(bm(s)) dβm(s)

+1

2

∫ t

0

∞∑m=1

(D2

2Φ(s, I(X(s))

)I(bm(s))

)I(bm(s)) ds

= Φ(0, X(0)

)+

∫ t

0

∂tΦ(s,X(s)

)ds+

∫ t

0

D2Φ(s,X(s)

)f(s) ds

+∞∑m=1

∫ t

0

D2Φ(s,X(s)

)bm(s) dβm(s)

+1

2

∫ t

0

∞∑m=1

(D2

2Φ(s,X(s)

)bm(s)

)bm(s) ds.

(2) We next extend (4.2) to the case where X is represented by arbitrary adapted

processes f and b1, . . . , bM . By the assumption of the theorem we have bm ∈L0(Ω;L2([0, T ];Lr(U1))) ∩ L0(Ω;Lrγ,T ) for any m = 1, . . . ,M .


Define the sequence (Xn)∞n=1 in L0(Ω;C([0, T ];Lr(U1))) by

Xn(t) := x0,n +

∫ t

0

fn ds+M∑m=1

∫ t

0

b(m)n dβm,

where (x0,n)∞n=1 is a sequence of F0-measurable simple functions with x0 =

limn→∞ x0,n almost surely, and (fn)∞n=1 and (b(m)n )∞n=1 are taken from Lemma 4.15.

By Theorem 3.40 we have X = limn→∞Xn in L0(Ω;C([0, T ];Lr(U1))). By pass-

ing to a subsequence we may assume that there exists a set Ω0 ⊆ Ω of full measure

such that

X = limn→∞

Xn in C([0, T ];Lr(U1)) on Ω0. (?1)

From this we deduce that

limn→∞

Φ(t,Xn(t)

)− Φ

(0, Xn(0)

)= Φ

(t,X(t)

)− Φ

(0, X(0)

)on Ω0.

For a continuous function Ψ: [0, T ]×Lr(U1)→ E, where E is some Banach space,

and ω ∈ Ω0 fixed, the set

∞⋃n=1

Ψ(s,Xn(s, ω)

): s ∈ [0, T ]

∪

Ψ(s,X(s, ω)

): s ∈ [0, T ]

is bounded. By applying this to the functions ∂tΦ, D2Φ and D2

2Φ, the random

variable

K = K(ω) := maxt∈[0,T ]

|∂tΦ

(t,X(t)

)|, ‖D2Φ

(t,X(t)

)‖, ‖D2

2Φ(t,X(t)

)‖,

|∂tΦ(t,Xn(t)

)|, ‖D2Φ

(t,Xn(t)

)‖, ‖D2

2Φ(t,Xn(t)

)‖

is finite for each fixed ω ∈ Ω0. Thus, by (?1) and the dominated convergence

theorem, we obtain on Ω0

limn→∞

∫ t

0

∂tΦ(s,Xn(s)

)ds =

∫ t

0

∂tΦ(s,X(s)

)ds.

Since

‖D2Φ(·, X)f‖L1([0,T ]) ≤ K‖f‖L1([0,T ];Lr(U1)) <∞,

(?1) and the dominated convergence theorem yield

limn→∞

‖D2Φ(·, Xn)f −D2Φ(·, X)f‖L1([0,T ]) = 0.

And by Lemma 4.15 we have

limn→∞

‖D2Φ(·, Xn)fn −D2Φ(·, Xn)f‖L1([0,T ]) ≤ limn→∞

K‖fn − f‖L1([0,T ];Lr(U1)) = 0.


Together these estimates give limn→∞ ‖D2Φ(·, Xn)fn − D2Φ(·, X)f‖L1([0,T ]) = 0,

and this implies that on Ω0

limn→∞

∫ t

0

D2Φ(s,Xn(s)

)fn(s) ds =

∫ t

0

D2Φ(s,X(s)

)f(s) ds.

Now fix an m ∈ 1, . . . ,M, and observe that bm(ω) ∈ L2([0, T ];Lr(U1)) for each

ω ∈ Ω0. Thus, similar as we just did, we obtain by Lemma 4.15, (?1), and the

dominated convergence theorem,

limn→∞

‖D2Φ(·, Xn)b(m)n −D2Φ(·, X)bm‖L2([0,T ])

≤ limn→∞

(‖D2Φ(·, Xn)(b(m)

n − bm)‖L2([0,T ]) +∥∥(D2Φ(·, Xn)−D2Φ(·, X)

)bm∥∥L2([0,T ])

)≤ lim

n→∞

(K‖b(m)

n − bm‖L2([0,T ];Lr(U1)) +∥∥(D2Φ(·, Xn)−D2Φ(·, X)

)bm∥∥L2([0,T ])

)= 0.

By Theorem 3.40 (for the R-valued case), this implies

limn→∞

∫ t

0

D2Φ(s,Xn(s)

)b(m)n (s) dβm(s) =

∫ t

0

D2Φ(s,X(s)

)bm(s) dβm(s)

on Ω0. For each summand in the last term in (4.2) we have∥∥(D22Φ(·, X)bm

)bm −

(D2

2Φ(·, Xn)b(m)n

)b(m)n

∥∥L1([0,T ])

≤∥∥(D2

2Φ(·, X)bm)bm −

(D2

2Φ(·, Xn)bm)bm∥∥L1([0,T ])

+∥∥(D2

2Φ(·, Xn)bm)bm −

(D2

2Φ(·, Xn)b(m)n

)b(m)n

∥∥L1([0,T ])

.

Using the Cauchy-Schwarz inequality, we get

‖D22

(Φ(·, X)bm

)bm‖L1([0,T ]) ≤ K‖bm‖2

L2([0,T ];Lr(U1)) <∞,

and thus, by (?1) and the dominated convergence theorem, the first term tends

to 0 on Ω0 . For the second term, by the Cauchy-Schwarz inequality, we have∥∥(D22Φ(·, Xn)bm

)bm −

(D2

2Φ(·, Xn)b(m)n

)b(m)n

∥∥L1([0,T ])

≤∥∥(D2

2Φ(·, Xn)bm)bm −

(D2

2Φ(·, Xn)b(m)n

)bm∥∥L1([0,T ])

+∥∥(D2

2Φ(·, Xn)b(m)n

)bm −

(D2

2Φ(·, Xn)b(m)n

)b(m)n

∥∥L1([0,T ])

≤ K‖b(m)n − bm‖L2([0,T ];Lr(U1))‖bm‖L2([0,T ];Lr(U1))

+K‖b(m)n − bm‖L2([0,T ];Lr(U1))‖b(m)

n ‖L2([0,T ];Lr(U1)),

which tends to 0 on Ω0 by Lemma 4.15. Finally, putting all these estimates

together and applying (1) to each Xn, we have shown that (4.2) is true for

arbitrary adapted processes f and b1, . . . , bM .


(3) We next assume that X is an arbitrary Lr(U1)-valued Ito process given by

dX = f dt+∑∞

m=1 bm dβm, where (βm)∞m=1 is a sequence of independent Brownian

motions. For M ∈ N we define

XM := x0 +

∫ ·0

f dt+M∑m=1

∫ ·0

bm dβm.

Since∥∥(∫ T

0

∑∞m=1 |bm|2 dt

) 12∥∥r<∞, Theorem 4.7 implies that X = limM→∞XM

in L0(Ω;C([0, T ];Lr(U1))). Hence, by passing to a subsequence, we may choose

Ω0 ⊆ Ω of full measure such that

X = limM→∞

XM in C([0, T ];Lr(U1)) on Ω0. (?2)

Let K : Ω→ R be as in the second part of this proof. Then, by (?2), we have

limM→∞

Φ(t,XM(t)

)− Φ

(0, XM(0)

)= Φ

(t,X(t)

)− Φ

(0, X(0)

)on Ω0.

Additionally, by (?2) and the dominated convergence theorem, we obtain on Ω0

limM→∞

∫ t

0

∂tΦ(s,XM(s)

)ds =

∫ t

0

∂tΦ(s,X(s)

)ds,

limM→∞

∫ t

0

D2Φ(s,XM(s)

)f(s) ds =

∫ t

0

D2Φ(s,X(s)

)f(s) ds.

Next, we take a look at the stochastic integral terms. Here, we have

limM→∞

∫ T

0

∞∑m=M+1

∣∣D2Φ(s,XM(s)

)bm∣∣2 ds ≤ lim

M→∞K2

∫ T

0

∞∑m=M+1

‖bm(s)‖2r ds = 0,

by the assumption we made to the sequence (bm)∞m=1. In addition, by (?2) and

the dominated convergence theorem, we obtain

limM→∞

∫ T

0

∞∑m=1

∣∣∣(D2Φ(s,XM(s)

)−D2Φ

(s,X(s)

))bm

∣∣∣2 ds = 0.

Let (bm)∞m=1 be given by bm = bm for 1 ≤ m ≤M and bm = 0 for m > M . Then,

by the previous estimates, we obtain

limM→∞

∫ T

0

∞∑m=1

∣∣D2Φ(s,XM(s)

)bm −D2Φ

(s,X(s)

)bm∣∣2 ds

≤ limM→∞

(∫ T

0

∞∑m=M+1

∣∣D2Φ(s,XM(s)

)bm∣∣2 ds

+

∫ T

0

∞∑m=1

∣∣∣(D2Φ(s,XM(s)

)−D2Φ

(s,X(s)

))bm

∣∣∣2 ds)

= 0.


By Theorem 4.7, this leads to

limM→∞

M∑m=1

∫ t

0

D2Φ(s,XM(s)

)bm dβm =

∞∑m=1

∫ t

0

D2Φ(s,X(s)

)bm dβm on Ω0.

For the last term in (4.2), we have

∥∥∥ ∞∑m=1

(D2

2Φ(·, X)bm)bm −

M∑m=1

(D2

2Φ(·, XM)bm)bm

∥∥∥L1([0,T ])

≤∥∥∥ ∞∑m=1

(D2

2Φ(·, X)bm)bm −

∞∑m=1

(D2

2Φ(·, XM)bm)bm

∥∥∥L1([0,T ])

+∥∥∥ ∞∑m=1

(D2

2Φ(·, XM)bm)bm −

M∑m=1

(D2

2Φ(·, XM)bm)bm

∥∥∥L1([0,T ])

.

Again, by (?2) and the dominated convergence theorem, the first term tends to

0 on Ω0. For the second term, we have

∥∥∥ ∞∑m=1

(D2

2Φ(·, XM)bm)bm −

M∑m=1

(D2

2Φ(·, XM)bm)bm

∥∥∥L1([0,T ])

=∥∥∥ ∞∑m=M+1

(D2

2Φ(·, XM)bm)bm

∥∥∥L1([0,T ])

≤ K∥∥∥( ∞∑

m=M+1

‖bm‖2r

) 12∥∥∥2

L2([0,T ]),

which also tends to 0 on Ω0, by the assumption we made to the sequence (bm)∞m=1.

Again, by collecting all estimates we have established in this part, and by apply-

ing (2) to each XM , we get the desired result.

(4) Finally, we show (4.2) as posted in the theorem, i.e., Φ is now an Ls(U2)-

valued function. For all g ∈ Ls′(U2) we almost surely have∫ T

0

∞∑m=1

∣∣∣⟨D2Φ(s,X(s)

)bm(s), g

⟩∣∣∣2 ds

≤ ‖g‖2Ls′ (U2)

maxt∈[0,T ]

∥∥D2Φ(t,X(t)

)∥∥2∫ T

0

∞∑m=1

‖bm(s)‖2r ds <∞,

since t 7→ D2Φ(t,X(t)

)is continuous. Now fix g ∈ Ls′(U2). Observe that differ-

entiating Φ and applying 〈·, g〉 are commuting operations.


With this in mind, part (3) applied to the R-valued function 〈Φ, g〉 leads to

〈Φ(t,X(t)

), g〉 = 〈Φ

(0, X(0)

), g〉+

⟨∫ t

0

∂tΦ(s,X(s)

)ds, g

⟩+⟨∫ t

0

D2Φ(s,X(s)

)f(s) ds, g

⟩+∞∑m=1

∫ t

0

〈D2Φ(s,X(s)

)bm(s), g〉 dβm(s)

+⟨1

2

∫ t

0

∞∑m=1

(D2

2Φ(s,X(s)

)bm(s)

)bm(s) ds, g

⟩.

An application of Theorem 4.8 to the pathwise continuous process

Y := Φ(·, X(·)

)− Φ

(0, X(0)

)−∫ ·

0

∂tΦ(s,X(s)

)ds−

∫ ·0

D2Φ(s,X(s)

)f(s) ds

− 1

2

∫ ·0

∞∑m=1

(D2

2Φ(s,X(s)

)bm(s)

)bm(s) ds

shows that∑∞

m=1

∫ t0D2Φ

(s,X(s)

)bm(s) dβm(s) is well-defined and equals almost

surely Y , which means that (4.2) holds.

Corollary 4.18. Let 1 < r1 < ∞, and define r2 := r′1. For i = 1, 2, let Xi

be an Lri(U)-valued Ito process given by dXi = fi dt+∑∞

m=1 bm,i dβm, which

satisfies the assumptions of Theorem 4.16. Then, almost surely we have for

all t ∈ [0, T ]

〈X1(t), X2(t)〉 = 〈X1(0), X2(0)〉+

∫ t

0

〈X1(s), f2(s)〉+ 〈f1(s), X2(s)〉 ds

+∞∑m=1

∫ t

0

〈X1(s), bm,2(s)〉+ 〈bm,1(s), X2(s)〉 dβm(s) (4.3)

+

∫ t

0

∞∑m=1

〈bm,1(s), bm,2(s)〉 ds.

Proof. Define Φ: [0, T ] × Lr1(U) × Lr′1(U) → R by Φ(t, x, y) := 〈x, y〉 (and

observe that Theorem 4.16 could be proved for these functions in the exact same

way with the same formula). Then Φ is of class C1,2 with

∂tΦ(t, x, y) = 0,

D2Φ(t, x, y) : Lr1(U)× Lr′1(U)→ R, (h, h′) 7→ 〈h, y〉+ 〈x, h′〉 ,D2

2Φ(t, x, y) : Lr1(U)× Lr′1(U)→(Lr1(U)× Lr′1(U)

)∗, (h, h′) 7→ 〈h, ·〉+ 〈·, h′〉 .

And finally, an application of Theorem 4.16 gives (4.3).


4.3 An Infinite Dimensional Version of the Ge-

ometric Brownian Motion

The final section is dedicated to an application of the theory developed in this

thesis to a stochastic partial differential equation. In this context, we will use

some notions and results from the theory of semigroups1. More precisely, we want

to study the problem

U(t) = u0 +

∫ t

0

AU(s) ds+N∑n=1

∫ t

0

BnU(s) dβn(s), t ∈ [0, T ], (4.4)

which we also write as

dU(t) = AU(t) dt+N∑n=1

BnU(t) dβn(t), t ∈ [0, T ],

U(0) = u0.

Here, the processes βn =(βn(t)

)t∈[0,T ]

are independent Brownian motions de-

fined on some probability space (Ω,A ,P) and adapted to some filtration F =

(Ft)t∈[0,T ]. We assume that the initial random variable u0 : Ω → Lr(U) is

F0-measurable, and regarding the operators A : D(A) ⊆ Lr(U) → Lr(U) and

Bn : D(Bn) ⊆ Lr(U)→ Lr(U) we make the following hypotheses:

(H1) The operator A is closed and densely defined.

(H2) The operators Bn generate commuting C0-groups Gn =(Gn(t)

)t∈R on

Lr(U), which also commute with A.

(H3) We have D(A) ⊆⋂Nn=1D(B2

n).

Defining D(C) := D(A) and C := A− 12

∑Nn=1B

2n, we further assume:

(H4) The operator C generates a C0-semigroup S =(S(t)

)t≥0

on Lr(U), and

u0 ∈ D(C) almost surely.

We call an Lr(U)-valued process U =(U(t)

)t∈[0,T ]

a strong solution of (4.4) on the

interval [0, T ] if U ∈ C([0, T ];Lr(U)) almost surely, U(0) = u0, and the following

conditions are satisfied:

(1) For almost all ω ∈ Ω, U(t, ω) ∈ D(A) for almost all t ∈ [0, T ] and the path

t 7→ AU(t, ω) belongs to L1([0, T ];Lr(U)).

(2) For each n = 1, . . . , N , the process BnU is an element of L0F(Ω;Lrγ,T ).

(3) Almost surely, U solves (4.4) on [0, T ].

1For more information on this topic we refer to [4] and [13].

4.3 An Infinite Dimensional Version of the Geometric Brownian Motion 109

Lemma 4.19. If (H2) holds, then D′ :=⋂Nn=1D(B∗2n ) is dense in Lr

′(U).

Proof. Since Lr(U) is reflexive, an application of the Hahn-Banach theorem

says that a linear subspace E ⊆ Lr′(U) is dense in Lr

′(U) if and only if for all

f ∈ Lr(U)\0 there exists a g ∈ E with 〈f, g〉 6= 0. Thus fix an f ∈ Lr(U)\0and some λ ∈

⋂Nn=1 %(Bn). Take f :=

∏Nn=1R(λ,Bn)2f . Since f 6= 0, we

can find a g ∈ Lr′(U) such that 〈f , g〉 6= 0. By (H2), the resolvents R(λ,B∗n)

commute, which implies that g :=∏N

n=1R(λ,B∗n)2g ∈⋂Nn=1D(B∗2n ). Finally, by

construction, we have 〈f, g〉 = 〈f , g〉 6= 0, and this concludes the proof.

Theorem 4.20. Assuming (H1)-(H4), the problem (4.4) has a strong solu-

tion on [0, T ].

Proof. Define G : RN → B(Lr(U)) as

G(x) :=N∏n=1

Gn(xn)

and the process Gβ : Ω× [0, T ]→ B(Lr(U)) by

Gβ(ω, t) := G(β1(ω, t), . . . , βN(ω, t)

),

which is adapted and pathwise strongly continuous. Next, we introduce the

following pathwise problem:

V ′(t) = CV (t), t ∈ [0, T ],

V (0) = u0.(4.5)

By (H4), the unique solution of this problem is V (t) = S(t)u0. Moreover, almost

surely we have V ∈ C1([0, T ];Lr(U)) and V (t) ∈ D(C) = D(A) for all t ∈[0, T ]. Put U := GβV . By (H2) we then have U(t) ∈ D(A), and since almost

surely CV ∈ C([0, T ];Lr(U)) and since Gβ is strongly continuous, we have CU =

CGβV = GβCV ∈ C([0, T ];Lr(U)) ⊆ L1([0, T ];Lr(U)). Let λ ∈ %(C) ∩ R. Then

we have

B2nU = B2

nR(λ,C)(λI − C)GβV = λB2nR(λ,C)GβV −B2

nR(λ,C)CGβV,

which is almost surely continuous. Therefore, B2nU ∈ L1([0, T ];Lr(U)) almost

surely, and this implies that AU = CU + 12

∑Nn=1 B

2nU ∈ L1([0, T ];Lr(U)) almost

surely.


Let D′ as in the previous lemma, and let g ∈ D′ be fixed. The function Φ: RN →Lr′(U) defined by Φ(x) := G(x)∗g is twice continuously differentiable with

∂Φ

∂xn(x) = G(x)∗B∗ng and

∂2Φ

∂x2n

(x) = G(x)∗B∗2n g.

We next define the RN -valued Ito process Xβ by

Xβ(t) :=N∑n=1

∫ t

0

en dβn =(β1(t), . . . , βN(t)

),

where (en)Nn=1 is the standard basis of RN . Hence, we have Φ(Xβ(t)) = Gβ(t)∗g,

and Ito’s formula yield

Gβ(t)∗g = Gβ(0)∗g +N∑n=1

∫ t

0

Gβ(s)∗B∗ng dβn(s) +1

2

N∑n=1

∫ t

0

Gβ(s)∗B∗2n g ds.

Moreover, since V is the solution of (4.5), we have

V (t) = V (0) +

∫ t

0

CV (s) ds.

Thus, by applying (4.3) to the Ito processes V and G∗βg, we almost surely obtain

for all t ∈ [0, T ]

〈U(t), g〉 − 〈U(0), g〉 = 〈V (t), Gβ(t)∗g〉 − 〈V (0), Gβ(0)∗g〉

=

∫ t

0

1

2

N∑n=1

⟨V (s), Gβ(s)∗B∗2n g

⟩+ 〈CV (s), Gβ(s)∗g〉 ds

+N∑n=1

∫ t

0

〈V (s), Gβ(s)∗B∗ng〉 dβn(s)

=

∫ t

0

〈AGβ(s)V (s), g〉 ds+N∑n=1

∫ t

0

〈BnGβ(s)V (s), g〉 dβn(s)

=

∫ t

0

〈AU(s), g〉 ds+N∑n=1

∫ t

0

〈BnU(s), g〉 dβn(s),

where we used (H2) and the definition of C. Since U = GβV is almost surely

continuous and since AU ∈ L1([0, T ];Lr(U)), the process

Y := U − U(0)−∫ ·

0

AU(s) ds

is almost surely continuous.

4.3 An Infinite Dimensional Version of the Geometric Brownian Motion 111

Finally, thanks to Lemma 4.19, we are now able to apply Remark 4.9, which

yields that BnU ∈ L0(Ω;Lrγ,T ) for each n = 1, . . . , N , and also implies that we

almost surely have

U(t) = u0 +

∫ t

0

AU(s) ds+N∑n=1

∫ t

0

BnU(s) dβn(s) for all t ∈ [0, T ].

Appendix A

Appendix

A.1 Integration in Banach Spaces

In this section we want to give a brief introduction to the Bochner integral, which

is a generalization of the Lebesgue integral to the Banach space valued setting.

First we are going to show some measurability results and after that we will con-

struct the Bochner integral.

A.1.1 Measurability

In what follows, let (A,A ) be a measurable space, E be an arbitrary real Banach

space, and B(E) be the Borel σ-algebra of E. We call a function f : A → E

A -simple if it is of the form f =∑N

n=1 1Anxn with An ∈ A and xn ∈ E for all

1 ≤ n ≤ N .

Definition A.1. A function f : A→ E is called A -measurable if

f−1(B) ∈ A for all B ∈ B(E),

and strongly A -measurable if there exists a sequence of A -simple functions

fn : A→ E such that

limn→∞

fn = f pointwise on A.

Next, we give a characterization of strong A -measurability of E-valued functions

(cf. [3, Chapter II.1, Theorem 2]). Here we call a function f : A → E separably

valued if there exists a separable closed subspace E0 of E such that f(a) ∈ E0

for all a ∈ A.

114 Appendix

Theorem A.2 (Pettis measurability theorem I). For any function

f : A→ E the following assertions are equivalent:

(1) f is strongly A -measurable;

(2) f is separably valued and 〈f, x∗〉 is A -measurable for all x∗ ∈ E∗.

As a consequence we obtain the following result.

Proposition A.3. For a function f : A → E the following assertions are

equivalent:

(1) f is strongly A -measurable;

(2) f is A -measurable and separably valued.

Thus, if E is separable, then an E-valued function f is strongly A -measurable if

and only if it is A -measurable.

Proof. (1) ⇒ (2): By Theorem A.2, f is separably valued. To show A -

measurability of f it suffices to prove that f−1(U) ∈ A for any open subset

U ⊆ E. Let U be open and choose a sequence of A -simple functions fn converg-

ing pointwise to f . Since U is open, we then get

f−1(U) ⊆∞⋃n=1

∞⋂k=n

f−1k (U).

For the converse inclusion, let Ur := x ∈ U : d(x, UC) > r. Then U =⋃∞m=1 U 1

m

and for each fixed m ∈ N we have

∞⋃n=1

∞⋂k=n

f−1k (U 1

m) ⊆ f−1(U).

Putting both estimates together, we obtain

f−1(U) =∞⋃m=1

∞⋃n=1

∞⋂k=n

f−1k (U 1

m).

Since f−1k (U 1

m) ∈ A , it follows that f is A -measurable.

(2) ⇒ (1): By the A -measurability of f we infer that 〈f, x∗〉 is A -measurable

for all x∗ ∈ E∗ and the result now follows from Theorem A.2.

A.1 Integration in Banach Spaces 115

So far we have considered measurability properties of E-valued functions defined

on a measurable space (A,A ). Next we consider functions defined on a σ-finite

measure space (A,A , ν). A function f : A → E is called ν-simple if it is of the

form f =∑N

n=1 1Anxn with xn ∈ E, and the sets An ∈ A satisfy ν(An) <∞.

Definition A.4. A function f : A → E is called strongly ν-measurable if

there exists a sequence of ν-simple functions fn : A→ E such that

limn→∞

fn = f ν-almost everywhere.

Using the σ-finiteness of ν, we can show that every strongly A -measurable func-

tion f is strongly ν-measurable. For this purpose, let (fn)∞n=1 be a sequence of

A -simple functions converging to f pointwise, and A1 ⊆ A2 ⊆ . . . in A with

A =⋃∞n=1 An and ν(An) < ∞ for each n ∈ N. Then also limn→∞ 1Anfn = f

pointwise and each 1Anfn is ν-simple.

A similar result is also true for the converse direction. Here we call two functions

ν-versions of each other if they agree ν-almost everywhere.

Proposition A.5. For a function f : A → E the following assertions are

equivalent:

(1) f is strongly ν-measurable;

(2) f has a ν-version which is strongly A -measurable.

Proof. (1) ⇒ (2): Let (fn)∞n=1 be a sequence of ν-simple functions converging

to f outside the null set N ∈ A . Then the functions 1NCfn are A -simple and

satisfy limn→∞ 1NCfn = 1NCf pointwise on A. Thus 1NCf is a strongly A -

measurable ν-version of f .

(2) ⇒ (1): Let f be a strongly A -measurable ν-version of f and (fn)∞n=1 be a

sequence of A -simple functions converging to f . Then limn→∞ fn = f ν-almost

everywhere. Moreover, write A =⋃∞n=1 An with A1 ⊆ A2 ⊆ . . . ∈ A and

ν(An) < ∞ for all n ∈ N. By taking fn := 1An fn, we obtain a sequence of

ν-simple functions (fn)∞n=1 with limn→∞ fn = f ν-almost everywhere.

By combining this proposition and Theorem A.2, we get the following result.

116 Appendix

Theorem A.6 (Pettis measurability theorem II). For any function

f : A→ E the following assertions are equivalent:

(1) f is strongly ν-measurable;

(2) f has a ν-version f which is separably valued and 〈f , x∗〉 is A -measur-

able for all x∗ ∈ E∗.

As a consequence of this theorem, we get the following results (cf. [3, Chapter

II.1, Corollary 3]).

Corollary A.7. (1) The ν-almost everywhere limit of a sequence of strongly

ν-measurable E-valued functions is strongly ν-measurable.

(2) If f : A → E is strongly ν-measurable and Φ: E → F is continuous,

where F is another Banach space, then Φ f is strongly ν-measurable.

We finish the discussion of ν-measurability with a quite useful corollary (cf. [3,

Chapter II.2, Corollary 7]).

Corollary A.8. If f and g are strongly ν-measurable E-valued functions

with 〈f, x∗〉 = 〈g, x∗〉 ν-almost everywhere for every x∗ ∈ E∗, then f = g

ν-almost everywhere.

A.1.2 The Bochner Integral

We next concentrate on the construction of the Bochner integral and give a short

introduction of the Lebesgue-Bochner spaces.

Definition A.9. A function f : A → E is called ν-Bochner integrable if

there exists a sequence of ν-simple functions fn : A → E such that the fol-

lowing two conditions hold:

(1) limn→∞ fn = f ν-almost everywhere (i.e., f is strongly ν-measurable);

(2) limn→∞∫A‖fn − f‖E dν = 0.

A.1 Integration in Banach Spaces 117

From the definition it is easy to see that every ν-simple function is ν-Bochner

integrable. For f =∑N

n=1 1Anxn we put

∫A

f dν :=N∑n=1

ν(An)xn.

Similar to the Lebesgue integral, we can check that this definition is independent

of the representation of f (and that implies that the integral is linear) . If f is

ν-Bochner integrable, the limit∫A

f dν := limn→∞

∫A

fn dν

exists in E and is called the Bochner integral of f with respect to ν. It is rou-

tine to check that this definition is independent of the approximating sequence

of f . Moreover, if f is ν-Bochner integrable and g is a ν-version of f , then g is

ν-Bochner integrable, and the Bochner integrals of f and g agree.

In the next proposition we collect some properties regarding ν-Bochner integra-

bility (cf. [3, Chapter II.2, Theorem 2,4 and 6]).

Proposition A.10. Let f : A → E be strongly ν-measurable. Then the

following assertions hold:

(1) f is ν-Bochner integrable if and only if∫A‖f‖E dν <∞. In this case,

we have ∥∥∥∫A

f dν∥∥∥E≤∫A

‖f‖E dν.

(2) If f is ν-Bochner integrable and T : E → F is a bounded linear opera-

tor, where F is another Banach space, then Tf : A → F is ν-Bochner

integrable and

T

∫A

f dν =

∫A

Tf dν.

Especially, we have ⟨∫A

f dν, x∗⟩

=

∫A

〈f, x∗〉 dν

for all x∗ ∈ E∗.

Next, we give an analogue of the dominated convergence theorem (cf. [3, Chapter

II.2, Theorem 3]).

118 Appendix

Proposition A.11 (Dominated convergence theorem). Let (fn)∞n=1 be

a sequence of E-valued ν-Bochner integrable functions. Assume that there

exist a function f : A → E and a ν-Bochner integrable function g : A → Rsuch that:

(1) limn→∞ fn = f ν-almost everywhere;

(2) ‖fn‖E ≤ |g| ν-almost everywhere.

Then f is ν-Bochner integrable, and we have

limn→∞

∫A

‖fn − f‖E dν = 0.

In particular, we have

limn→∞

∫A

fn dν =

∫A

f dν.

We finish this section with an introduction of the Lp spaces for Banach space

valued functions, the so called Lebesgue-Bochner spaces. For 1 ≤ p <∞ we define

Lp(A;E) as the linear space of all equivalence classes of strongly ν-measurable

functions f : A→ E satisfying ∫A

‖f‖pE dν <∞,

identifying functions which are equal ν-almost everywhere. As in the scalar case,

we can show that the space Lp(A;E) endowed with the norm

‖f‖p :=(∫

A

‖f‖pE dν) 1p

is a Banach space. By the definition of the Bochner integral and Proposition

A.10 (1), it is easy to see that ν-simple functions are dense in Lp(A;E).

We define L∞(A;E) as the linear space of all equivalence classes of strongly

ν-measurable functions f : A→ E such that there exists an r ≥ 0 with

ν(‖f‖E > r

)= 0.

Endowed with the norm

‖f‖∞ := infr ≥ 0: ν

(‖f‖E > r

)= 0,

the space L∞(A;E) is a Banach space.

A.2 Gaussian Random Variables 119

Remark A.12. In the case of equivalence classes, we will just say that f is

strongly measurable if f has a strongly ν-measurable representative (in this case,

every representative is strongly ν-measurable and by Proposition A.5 there even

exists a representative which is strongly A -measurable). So, in what follows,

we will omit the prefix ’ν−’ from our terminology if no confusion can arise.

Especially, if E is separable, we will just say that a strongly measurable function

f is measurable (motivated by Proposition A.3).

A.2 Gaussian Random Variables

Let (Ω,A ,P) be a probability space and E be an arbitrary real Banach space. An

E-valued random variable is an E-valued strongly measurable function X : Ω→E. For E-valued random variables, the definitions from the R-valued case carries

over nearly verbatim. For example, the Bochner integral of an integrable random

variable X is called its mean value or expectation, and is denoted by

EX :=

∫Ω

X dP.

Moreover, the distribution of an E-valued random variable X is the Borel prob-

ability measure PX on E defined by

PX(B) := P(X ∈ B), B ∈ B(E).

Definition A.13. The Fourier transform of a Borel probability measure µ

on E is the function µ : E∗ → C defined by

µ(x∗) :=

∫E

exp(−i 〈x, x∗〉

)dµ(x).

The Fourier transform of a random variable X : Ω→ E is the Fourier trans-

form of its distribution PX .

Note that µ is well-defined since∣∣exp

(−i 〈x, x∗〉

)∣∣ = 1, which implies that the

integral is absolutely convergent. By a change of variable, the Fourier transform

of a random variable X on E is given by

X(x∗) = E exp(−i 〈X, x∗〉

)=

∫E

exp(−i 〈x, x∗〉

)dPX(x).

We next show a uniqueness result (cf. [12, Chapter IV, Theorem 3.1 and Lemma

5.2]).

120 Appendix

Theorem A.14. Let X1 and X2 be E-valued random variables who satisfy

X1(x∗) = X2(x∗) for all x∗ ∈ E∗.

Then X1 and X2 are identically distributed.

Next, we proceed with Gaussian random variables. An R-valued random variable

γ is called Gaussian if there exists a number q ≥ 0 such that its Fourier transform

is given by

E exp(−iξγ) = exp(−12qξ2), ξ ∈ R.

Note that this definition is consistent with the one given in Chapter 1.

An E-valued random variable X is said to be Gaussian if the R-valued random

variable 〈X, x∗〉 is Gaussian for all x∗ ∈ E∗. Based on the uniqueness theorem

of the Fourier transform, we can now show the following properties of E-valued

Gaussian random variables.

Proposition A.15. Let X and Y be independent and identically distributed

E-valued Gaussian random variables. Then U := 1√2(X + Y ) and V :=

1√2(X − Y ) are independent and have the same distribution as X and Y .

Proof. We have X(x∗) = Y (x∗) = exp(−1

2q(x∗)

), where q(x∗) = E 〈X, x∗〉2 =

E 〈Y, x∗〉2. By the independence of X and Y , we obtain

U(x∗) = E exp(−i 1√

2〈X, x∗〉

)E exp

(−i 1√

2〈Y, x∗〉

)= exp

(−1

4q(x∗)

)exp(−1

4q(x∗)

)= exp

(−1

2q(x∗)

).

So, by Theorem A.14, U has the same distribution as X and Y . By a similar

computation we get the same result for V and the independence of U and V .

As a further application we prove next that if the E-valued random variables

X1, . . . , XN are jointly Gaussian, that is, if the EN -valued random variable X :=

(X1, . . . , XN) is Gaussian, then X1, . . . , XN are independent if and only if they

are uncorrelated in the sense that

E〈Xm, x∗〉〈Xn, y

∗〉 = 0, for all x∗, y∗ ∈ E∗ and m 6= n.

A.2 Gaussian Random Variables 121

Proposition A.16. LetX1, . . . , XN be E-valued random variables such that

the EN -valued random variable (X1, . . . , XN) is Gaussian. Then the following

assertions are equivalent:

(1) X1, . . . , XN are independent;

(2) X1, . . . , XN are uncorrelated.

Proof. For the R-valued case see [7, Lemma 13.1] or [9, Satz 7.33].

(1)⇒ (2): SinceX1, . . . , XN are independent, it follows that the random variables

〈X1, x∗1〉, . . . , 〈XN , x

∗N〉 are independent for any x∗1, . . . , x

∗N ∈ E∗. The implication

therefore follows from the corresponding implication in the R-valued case.

(2) ⇒ (1): Note that(〈X1, x

∗1〉, . . . , 〈XN , x

∗N〉)

is again an RN -valued Gaussian

random variable for any x∗1, . . . , x∗N ∈ E∗. So, by (2) and the foregoing remark,

the R-valued random variables 〈X1, x∗1〉, . . . , 〈XN , x

∗N〉 are independent. Now we

obtain

P(X1,...,XN )(x∗1, . . . , x

∗N) = E exp

(−i

N∑n=1

〈Xn, x∗n〉)

=N∏n=1

E exp(−i 〈Xn, x

∗n〉)

=N∏n=1

PXn(x∗n) = PX1 ⊗ . . .⊗ PXN (x∗1, . . . , x∗N).

Thus, Theorem A.14 implies P(X1,...,XN ) = PX1 ⊗ . . . ⊗ PXN , which shows that

X1, . . . , XN are independent.

Finally, the next result shows that limits of Gaussian random variables are again

Gaussian.

Proposition A.17. If (Xn)∞n=1 is a sequence of E-valued Gaussian random

variables and X is a random variable with

limn→∞

〈Xn, x∗〉 = 〈X, x∗〉 in probability for all x∗ ∈ E∗,

then X is Gaussian.

122 Appendix

Proof. Let x∗ ∈ E∗ be fixed. By passing to an appropriate subsequence, we have

limk→∞ 〈Xnk , x∗〉 = 〈X, x∗〉 almost surely. Therefore, the dominated convergence

theorem implies

E exp(−iξ 〈X, x∗〉

)= lim

k→∞E exp

(−iξ〈Xnk , x

∗〉)

= limk→∞

exp(−1

2ξ2qnk(x

∗)),

using that Xnk is Gaussian for each k ∈ N. From this we infer that the limit

q(x∗) := limk→∞ qnk(x∗) exists (observe that

((qnk(x

∗))∞k=1

is a non-negative,

bounded sequence). This leads to

E exp(−iξ〈X, x∗〉

)= exp

(−1

2ξ2q(x∗)

),

and Theorem A.14 finally yield that 〈X, x∗〉 is Gaussian.

A.3 Conditional Expectations and Martingales

Let (Ω,A ,P) be a probability space and G be a sub-σ-algebra of A . For

1 ≤ p ≤ ∞ we denote by Lp(Ω,G ) the subspace of all ξ ∈ Lp(Ω) having a G -

measurable representative. Note that Lp(Ω,G ) is a closed subspace of Lp(Ω). We

next want to show that Lp(Ω,G ) is the range of a contractive projection in Lp(Ω).

For p = 2 we have the orthogonal decomposition

L2(Ω) = L2(Ω,G )⊕ L2(Ω,G )⊥,

and we can choose the orthogonal projection PG onto L2(Ω,G ). As everyone

does, we also write

E[ξ|G ] := PG ξ, ξ ∈ L2(Ω),

and call E[ξ|G ] the conditional expectation of ξ with respect to G . Note that

E[ξ|G ] is an element of L2(Ω,G ) and therefore an equivalence class of random

variables.

Lemma A.18. For all ξ ∈ L2(Ω) and G ∈ G we have∫G

E[ξ|G ] dP =

∫G

ξ dP.

Proof. Let G ∈ G . Since 1G ∈ L2(Ω,G ) and ξ − E[ξ|G ] ∈ L2(Ω,G )⊥, we have∫Ω

1G(ξ − E[ξ|G ]

)dP = 0,

which gives the desired identity.

A.3 Conditional Expectations and Martingales 123

As a consequence, we get the following properties for ξ ∈ L2(Ω):

(1) If ξ ≥ 0 almost surely, then E[ξ|G ] ≥ 0 almost surely.

(2) By taking G = Ω in the previous lemma, we get

E(E[ξ|G ]

)= Eξ.

(3) Let ξ+ and ξ− be the positive and negative part of ξ, respectively. By (2)

we then have

E∣∣E[ξ|G ]

∣∣ ≤ E(E[ξ+|G ] + E[ξ−|G ]

)= E

(E[|ξ|∣∣G ]) = E|ξ|,

which shows that the map ξ 7→ E[ξ|G ] is L1-bounded.

Since L2(Ω) is dense in L1(Ω), we can extend the conditional expectation opera-

tor to a contractive projection on L1(Ω), which we also denote by E[ · |G ]. The

properties (1)–(3) then still hold for this projection.

Using this estimate and a conditional version of Jensen’s inequality, we get the

next theorem (cf. [3, Chapter V.1, Lemma 3] for more details).

Theorem A.19 (Lp-contractivity). For all 1 ≤ p ≤ ∞, the conditional

expectation operator extends to a contractive positive projection on Lp(Ω)

with range Lp(Ω,G ). For ξ ∈ Lp(Ω), the random variable E[ξ|G ] is the unique

element of Lp(Ω,G ) with the property that for all G ∈ G we have∫G

E[ξ|G ] dP =

∫G

ξ dP.

Our next aim is to extend these contractive operators from Lp(Ω) to Lp(Ω;E),

where E is some Banach space. For this purpose, fix 1 ≤ p < ∞. Then, by the

definition of the Lebesgue-Bochner spaces, the set

D :=X =

N∑n=1

ξnxn : ξn ∈ Lp(Ω), xn ∈ E, N ∈ N

is dense in Lp(Ω;E). Suppose next that T ∈ B(Lp(Ω)

). On the above set, we

then define a linear operator T ⊗ I by

(T ⊗ I)X =N∑n=1

Tξn · xn, X ∈ D.

For positive operators T we then have the following result (cf. [5]).

124 Appendix

Proposition A.20. If T is a positive operator on Lp(Ω), then T ⊗I extends

uniquely to a bounded operator on Lp(Ω;E), and we have ‖T ⊗ I‖ = ‖T‖.

Returning to the positive conditional expectation operator, we obtain the follow-

ing extension of Theorem A.19.

Theorem A.21 (Lp-contractivity). For 1 ≤ p ≤ ∞, the linear operator

E[ · |G ] ⊗ I extends uniquely to a contractive projection on Lp(Ω;E) with

range Lp(Ω,G ;E). For all X ∈ Lp(Ω;E), the random variable

E[X|G ] :=(E[ · |G ]⊗ I

)X

is the unique element of Lp(Ω,G ;E) with the property that for all G ∈ G we

have ∫G

E[X|G ] dP =

∫G

X dP.

Having defined the conditional expectation for random variables with values in a

Banach space, we next collect some of its properties.

(1) Let X ∈ L1(Ω;E). Then

E(E[X|G ]

)= EX. (A.1)

(2) If X ∈ L1(Ω,G ;E), then we almost surely have

E[X|G ] = X. (A.2)

Especially, if ξ ∈ Lp(Ω) and X ∈ Lp′(Ω,G ;E), then almost surely

E[ξX|G ] = E[ξ|G ]X, (A.3)

and if ξ ∈ Lp(Ω,G ) and X ∈ Lp′(Ω;E), then almost surely

E[ξX|G ] = ξ E[X|G ]. (A.4)

(3) If X ∈ L1(Ω;E) is independent of G (that is, X is independent of 1G for

all G ∈ G ), then almost surely

E[X|G ] = EX. (A.5)


(4) If X ∈ L1(Ω;E) and H is a sub-σ-algebra of G , then almost surely

E[E[X|G ]

∣∣H ]= E

[E[X|H ]

∣∣G ] = E[X|H ]. (A.6)

(5) Let H be a sub-σ-algebra of G and X ∈ L1(Ω;E). Assume that H is

independent of σ(X,G ) := σ(σ(X) ∪ G

)(i.e., the indicator functions 1H

and 1G are independent for all H ∈H and G ∈ σ(X,G )). Then we almost

surely have

E[X|G ,H ] := E[X|σ(G ∪H )] = E[X|G ]. (A.7)

(6) Let (Ω, A , P) be another probability space and G be a sub-σ-algebra of A .

Suppose that ξ ∈ L1(Ω) and X ∈ L1(Ω;E), then almost surely

E[ξX|G ⊗ G ] = E[ξ|G ] E[X|G ]. (A.8)

(7) Let X, Y ∈ L1(Ω;E) be independent and identically distributed. Then we

have

E[X − Y |X + Y ] := E[X − Y |σ(X + Y )] = 0. (A.9)

Proof. The assertions (1) - (4) follow directly from the uniqueness part of The-

orem A.21.

(5) We first consider an R-valued random variable ξ ≥ 0. We further assume that

Eξ > 0 (since otherwise ξ = 0 almost surely and there is nothing to prove) and

note that C := G ∩H : G ∈ G , H ∈ H is a generating system of σ(G ∪H ),

which is closed under taking finite intersections. By the independence of H and

σ(ξ,G ), we have for G ∩H ∈ C∫G∩H

E[ξ|G ,H ] dP =

∫G∩H

ξ dP = E[1G1Hξ]

= E1H E[1Gξ] = E1HE(E[1Gξ|G ]

)= E

(1G∩HE[ξ|G ]

)=

∫G∩H

E[ξ|G ] dP.

By Dynkin’s lemma, applied to the probability measures

µ1(C) :=1

Eξ

∫C

E[ξ|G ,H ] dP and µ2(C) :=1

Eξ

∫C

E[ξ|G ] dP,

it follows that µ1 = µ2 on σ(C) = σ(G ,H ), and this shows the desired estimate

for positive random variables ξ. For arbitrary R-valued random variables we

consider positive and negative parts separately. Finally, the vector-valued case

follows from the definition of the conditional expectation operator for ’simple’

functions and approximation.

126 Appendix

(6) By a similar argument as above, it suffices to prove the estimate for R-valued

random variables ρ. Therefore, we first consider ξ, ρ ≥ 0, and we may assume

that Eξ > 0 and Eρ > 0. Note that D := G1 × G2 : G1 ∈ G , G2 ∈ G is a

generating system of G ⊗ G , which is closed under finite intersections. Then, by

Fubini’s theorem, we obtain for G1 ×G2 ∈ D∫G1×G2

E[ξρ|G ⊗ G] d(P⊗ P) =

∫G1×G2

ξρ d(P⊗ P) =

∫G1

ξ dP∫G2

ρ dP

=

∫G1

E[ξ|G ] dP∫G2

E[ρ|G ] dP

=

∫G1×G2

E[ξ|G ]E[ρ|G ] d(P⊗ P).

As in (5), we apply Dynkin’s lemma with the probability measures

ν1(D) :=1

EξEρ

∫D

E[ξρ|G ⊗ G ] d(P⊗ P) and

ν2(D) :=1

EξEρ

∫D

E[ξ|G ]E[ρ|G ] d(P⊗ P)

to obtain ν1 = ν2 on G ⊗ G . This then shows the estimate for positive random

variables ξ and ρ. By splitting the random variables in positive and negative

parts and making a simple computation, we finally get

E[ξρ|G ⊗ G ] = E[ξ|G ]E[ρ|G ]

for arbitrary R-valued random variables ξ and ρ.

(7) Since X and Y are independent and identically distributed, we have

P(X,Y ) = PX ⊗ PY = PY ⊗ PX = P(Y,X).

Let A ∈ σ(X + Y ) be arbitrary. Then there exists a set B ∈ B(E) satisfying

A = (X + Y )−1(B), and this leads to∫A

X dP =

∫Ω

1AX dP =

∫E

1B(x, y)x dP(X,Y )

=

∫E

1B(y, x)y dP(Y,X) =

∫Ω

1AY dP =

∫A

Y dP.

From this we infer that

E[X|X + Y ] = E[Y |X + Y ] almost surely,

and the linearity of the conditional expectation operator finally shows the desired

estimate.


Now that the existence and properties of conditional expectations were discussed,

we next introduce E-valued martingales.

Let I be a partially ordered set. A filtration with index set I is a family (Fi)i∈Iof sub-σ-algebras of A such that Fi ⊆ Fj whenever i ≤ j. A family of E-

valued random variables is said to be adapted to the filtration (Fi)i∈I if each

Xi is strongly Fi-measurable (or more precisely, if each Xi has a strongly Fi-

measurable representative).

Definition A.22. A family (Mi)i∈I of integrable E-valued random variables

is an E-valued martingale with respect to a filtration (Fi)i∈I if it is adapted

to (Fi)i∈I and if for all i ≤ j we have

E[Mj|Fi] = Mi almost surely.

If in addition E‖Mi‖pE < ∞ for all i ∈ I and some 1 ≤ p < ∞, then we call

(Mi)i∈I an E-valued Lp martingale.

Example A.23 (Martingale transform). A sequence of R-valued random vari-

ables v = (vn)Nn=1 is said to be predictable with respect to a filtration (Fn)Nn=1 if vnis Fn−1-measurable for n = 1, . . . , N (with the understanding that F0 = ∅,Ω).If M = (Mn)Nn=1 is an E-valued martingale with respect to (Fn)Nn=1, then the

sequence v ∗M =((v ∗M)n

)Nn=1

defined by

(v ∗M)n :=n∑j=1

vj(Mj −Mj−1), n = 1, . . . , N,

is called the martingale transform of M by v (with the understanding that

M0 = 0).

If we further assume that each vn is bounded, then v ∗M is indeed a martingale

with respect to (Fn)Nn=1. In fact, by the boundedness the random variables

vj(Mj −Mj−1) are integrable and so also (v ∗M)n. Clearly, v ∗M is adapted to

(Fn)Nn=1. Moreover, by (A.4) and the Fn−1-measurability of (v ∗M)n−1 and vn,

we obtain

E[(v ∗M)n

∣∣Fn−1

]= (v ∗M)n−1 + vnE[Mn −Mn−1|Fn−1] = (v ∗M)n−1.

Finally, we collect some important results of E-valued Lp martingales, starting

with the famous martingale inequality by Doob (cf. [8, Proposition 4.1.1]).

128 Appendix

Theorem A.24 (Doob). Let 1 < p <∞, and let (Mn)Nn=1 be an E-valued

martingale with respect to (Fn)Nn=1. Define

M∗ : Ω→ [0,∞), M∗(ω) :=N

maxn=1‖Mn(ω)‖E.

If MN ∈ Lp(Ω;E), then

E|M∗|p ≤( p

p− 1

)pE‖MN‖pE.

Given a filtration (Fn)∞n=1 on (Ω,A ,P), we denote by F∞ the σ-algebra gen-

erated by (Fn)∞n=1, that is, F∞ is the smallest σ-algebra containing each sub-

σ-algebra Fn. The next result can be found in [3, Chapter V.2, Theorem 1 and 8].

Theorem A.25 (Martingale convergence theorem). Let 1 ≤ p < ∞,

and assume that X ∈ Lp(Ω;E). Then,

limn→∞

E[X|Fn] = E[X|F∞]

both in Lp(Ω;E) and almost surely.

Fix 1 ≤ r < ∞, and assume that (U,Σ, µ) is a σ-finite measure space and Σ is

countably generated. In this case we have the following result.

Lemma A.26. Let (Mn)Nn=1 be an Lr(U)-valued Lr martingale with respect

to a filtration (Fn)Nn=1. Then(Mn(u)

)Nn=1

is an R-valued Lr martingale with

respect to (Fn)Nn=1 for µ-almost all u ∈ U .

Proof. Since (U,Σ, µ) is a σ-finite measure space, U is a countable union of dis-

joint sets Uk ∈ Σ, k ∈ N, of finite measure. Take U (K) :=⋃Kk=1 Uk and let (UK

j )NKj=1

be a partition of U (K) in such a way that ΣK := σ(U

(K)j , j = 1, . . . , NK

)⊆ ΣM

whenever K ≤M , and Σ = σ(ΣK , K ∈ N). Next, we define for K ∈ N

PK := E[ · |ΣK ] : Lr(U)→ Lr(U).

Let f ∈ Lr(U). Since ΣK is generated by NK disjoint sets, PK has the following

representation

(PKf)(u) =

NK∑j=1

∫U

(K)j

f dµ

µ(U(K)j )

1U

(K)j

(u).


Let ε > 0. Since f ∈ Lr(U) and limn→∞ 1U(n)(u)f(u) = f(u) for all u ∈ U , the

dominated convergence theorem yield an N ∈ N such that

‖1U(K)f − f‖r <ε

6for all K ≥ N.

Moreover, for that N the martingale convergence theorem gives an M(N) ∈ Nsuch that ∥∥E[1U(N)f |ΣK ]− 1U(N)f

∥∥r<ε

6for all K ≥M(N),

since µ(U (N)) <∞. Thus for K ≥ maxN,M(N), we obtain∥∥E[f |ΣK ]− f∥∥r≤∥∥E[1U(K)f |ΣK ]− E[1U(N)f |ΣK ]

∥∥r

+∥∥E[1U(N)f |ΣK ]− 1U(N)f

∥∥r

+∥∥1U(N)f − 1U(K)f

∥∥r

+∥∥1U(K)f − f

∥∥r

≤ 2∥∥1U(N)f − 1U(K)f

∥∥r

+ε

3

≤ 2∥∥1U(N)f − f

∥∥r

+ 2∥∥1U(K)f − f

∥∥r

+ε

3

≤ ε,

where we used that E[f |ΣK ] = E[1U(K)f |ΣK ] by construction, and that the con-

ditional expectation operator is contractive (observe that µ(U (K)) <∞).

For each n = 1, . . . , N we next define the random variable M(k)n : Ω→ Lr(U) by

M (k)n (ω) := PkMn(ω) ω ∈ Ω, k ∈ N.

Let A ∈ Fn−1 be arbitrary. Since (Mn)Nn=1 is a martingale with respect to

(Fn)Nn=1, Theorem A.21 leads to∫A

Mn dP =

∫A

E[Mn|Fn−1] dP =

∫A

Mn−1 dP.

Therefore, by Fubini’s theorem, we obtain for each fixed k ∈ N∫A

M (k)n dP =

Nk∑j=1

1

µ(U(k)j )

1U

(k)j

∫A

∫U

(k)j

Mn dµ dP

=

Nk∑j=1

1

µ(U(k)j )

1U

(k)j

∫U

(k)j

∫A

Mn−1 dP dµ

=

∫A

Nk∑j=1

1

µ(U(k)j )

1U

(k)j

∫U

(k)j

Mn−1 dµ dP

=

∫A

M(k)n−1 dP.

130 Appendix

And again by Theorem A.21, this implies

E[M (k)n |Fn−1] = M

(k)n−1 almost surely.

Next, for any k ∈ N and each fixed u ∈ U , almost surely we have

E[M (k)n (u)|Fn−1] =

Nk∑j=1

1

µ(U(k)j )

E[∫

U(k)j

Mn dµ∣∣∣Fn−1

]1U

(k)j

(u)

= E[M (k)n |Fn−1](u) (4)

= M(k)n−1(u).

Having shown all these auxiliary results, we are now in the position to prove the

assertion claimed in the lemma. Fix an n ∈ 1, . . . , N. Since limk→∞M(k)n =

Mn and limk→∞M(k)n−1 = Mn−1 almost surely in Lr(U), and since Mn, Mn−1 ∈

Lr(Ω;Lr(U)), the dominated convergence theorem and Fubini’s theorem yield

limk→∞

M (k)n = Mn and lim

k→∞M

(k)n−1 = Mn−1 in Lr(Ω;Lr(U)) ' Lr(U ;Lr(Ω)).

Thus, we may find an appropriate subsequence (kj)j∈N and a µ-nullset U0 ⊆ U

such that

limj→∞

M (kj)n (u) = Mn(u) and lim

j→∞M

(kj)n−1(u) = Mn−1(u) in Lr(Ω) for all u /∈ U0.

Especially, Mn(u) ∈ Lr(Ω) for all u /∈ U0 and is therefore integrable. We next

observe that each M(k)n (u) is Fn-measurable by construction, which implies that

Mn(u) is Fn-measurable as a limit of Fn-measurable random variables for all

u /∈ U0. Finally, by (4), Theorem A.19, and the above convergence, we almost

surely get for all u /∈ U0

E[Mn(u)|Fn−1] = limj→∞

E[M (kj)n (u)|Fn−1] = lim

j→∞M

(kj)n−1(u) = Mn−1(u),

which concludes the proof.

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Erklarung

Hiermit versichere ich, dass ich die Arbeit selbstandig verfasst, keine anderen

als die angegebenen Quellen und Hilfsmittel benutzt, die wortlich oder inhaltlich

ubernommenen Stellen als solche kenntlich gemacht und die Satzung des Karls-

ruher Instituts fur Technologie zur Sicherung guter wissenschaftlicher Praxis in

der gultigen Fassung beachtet habe.

Karlsruhe, den 3. August 2012

Markus Antoni

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