Stochastic Integration and Ito’s Formulastaff.ustc.edu.cn/~wangran/Course/Hsu/Chapter 3... · Stochastic Integration and Ito’s Formula In this chapter we discuss Ito’s theory

CHAPTER 3

Stochastic Integration and Ito’s Formula

In this chapter we discuss Ito’s theory of stochastic integration. This is avast subject. However, our goal is rather modest: we will develop this the-ory only generally enough for later applications. We will discuss stochasticintegrals with respect to a Brownian motion and more generally with re-spect to a continuous local martingale. Instead of attempting to describethe largest possible class of integrand processes, we will only single out aclass of integrand processes sufficiently large for our later applications. Forexample, in the case of a continuous local martingale as an integrator, weonly restrict ourselves to continuous adapted integrand processes, a classof processes sufficiently large for most applications. This of course does notmean that integrands which are not continuous cannot be integrated withrespect to a continuous martingale. It is usually the case that when deal-ing with a discontinuous integrand, we can quickly decide then and therewhether the integral has a meaning. Since many excellent textbooks onstochastic integration are available (McKean [8], Ikeda and Watanabe [6],Chung and Williams [3], Oksendal [10], Karatzas and Shreve [7], to cite justa few), there is little motivation on the part of the author to go beyond whatwill be presented in this chapter.

1. Introduction

If A is a process of bounded variation and f : R×R is function suchthat s 7→ f (s, ω) is Borel measurable function, then∫ t

0f (s) dAs

can be interpreted as a pathwise integral if proper integrability conditionsare assumed and its value at ω ∈ Ω is the usual Lebesques–Stieljes integral∫ t

0f (s, ω) dAs(ω).

In theoretical and applied problems, there is an obvious need to make senseout of an integral of the form ∫ t

0f (s) dBs,

where B is a Brownian motion. As we know, Brownian motion samplepaths are not functions of bounded variation (see REMARK 7.2). For this

43

44 3. STOCHASTIC INTEGRATION AND ITO’S FORMULA

reason in general there is no easy and direct pathwise interpretation of theabove integral. However, in some special situation, a simple interpretationis possible. For example, if s 7→ f (s, ω) itself has bounded variation foreach ω, we can define the above integral by integration by parts:∫ t

0f (s) dBs = f (t)Bt −

∫ t

0Bs d f (s).

Such stochastic integrals are rather limited in its scope of application. Ito’stheory of stochastic integration greatly expands the class of integrand pro-cesses, thus making the theory into a powerful tool in pure and appliedmathematics.

We first define the integration of a step (and deterministic) process withrespect to a Brownian motion. Let

∆ : 0 = t0 < t1 · · · < tn = t

be a partition of [0, t]. If f is a step process:

f (s) = f j−1, tj−1 ≤ t < tj,

then the natural definition of its stochastic integral with respect to Browni-an motion B is ∫ t

0f (s) dBs =

n−1

∑j=1

f j−1

[Btj − Btj−1

].

Using the property of independent increments for Brownian motion, itssecond moment is given by

E

[∫ t

0f (s) dBs

]2

=n

∑i,j=1

fi1 f j−1E[(

Bti − Bti−1

) (Btj − Btj−1

)]=

n

∑j=1| f j−1|2E

[(Btj − Btj−1

)2]

=n

∑j=1| f j−1|2

(tj − tj−1

).

This gives

(1.1) E

∣∣∣∣∫ 1

0f (s) dBs

∣∣∣∣2 =n

∑j=1| f j−1|2(tj − tj−1) =

∫ t

0| f (s)|2 ds.

If f and g are two step functions, then

E|∫ t

0f (s) dBs −

∫ t

0g(s) dBs|2 =

∫ t

0| f (s)− g(s)|2 ds = ‖ f − g‖2

2.

For an arbitrary function f : [0, t]→ R, it is now clear that as long as thereis a sequence of step functions fn such that ‖ fn − f ‖2 → 0, we can define

1. INTRODUCTION 45

the stochastic integral as the limit∫ t

0f (s) dBs = lim

n→∞

∫ t

0fn(s) dBs.

It is well know from real analysis that this approximation property is sharedby all Borel measurable functions f such that ‖ f ‖2 < ∞. We have thusgreatly enlarged the space of deterministic functions which can be inte-grated with respect to a Brownian motion.

The key observation in Ito’s theory of stochastic integration is that wecan pass to a random step process as long as f j ∈ Ftj . Note that now theintegrand has the form

f (s) = f j−1, tj−1 ≤ s < tj, f j−1 ∈ Ftj−1 ,

Namely, in the time interval [tj−1, tj], the integrand is measurable with re-spect to the σ-algebra at the left endpoint of the time interval. For this rea-son above argument still applies as long as we replace the equality (1.1) bythe more general equality

E

∣∣∣∣∫ 1

0f (s) dBs

∣∣∣∣2 = En

∑j=1| f j−1|2(tj − tj−1) = E

∫ t

0| f (s)|2 ds.

The key step in establishing this equality is the vanishing of the off diagonalterm

E[

fi−1 f j−1(Bti − Bti1)(Btj − Btj−1)]= 0.

This holds because if i < j, then by conditioning on Ftj−1 , we have

E[

fi−1 f j−1(Bti − Bti1)(Btj − Btj−1)|Ftj−1

]= fi−1 f j−1(Bti − Bti1)E

[Btj − Btj−1 |Ftj−1

]= 0.

For the diagonal term, by conditioning on F tj− 1 again, we have

E[| fi−1|2(Bti − Bti−1)

2] = E| fi−1|2(ti − ti−1).

An important property of stochastic integrals is that

Mt =∫ t

0f (s) dBs

is a continuous martingale. Let us see why this is so if the integrand processis a step process

f (s) = f j−1, tj−1 < s ≤ tj, f j−1 ∈ L2(Ω, Ftj−1 , P).

We can write

Mt =∫ t

0f (s) dBs = ∑

nf j−1

(Btj ∧ t− Btj−1∧t

).


Note that sum has only finitely many terms for each fixed t. The processM has continuous sample paths because Brownian motion does. Supposethat s < t. If s and t are not among the points tj we may simply insert theminto the sequence. Now s and t are among the sequence we have

Mt −Ms = ∑s≤tj−1≤t

f j−1

(Btj − Btj−1

).

The general term in the sum vanishes after we condition it on Ftj−1 , henceit also vanishes after we condition on Fs because Fs ⊂ Ftj−1 . It followsthat

E [Mt −Ms|Fs] = 0

and Mt is a continuous martingale.We now calculate the quadratic variation process for the continuous

martingale

Mt =∫ t

0f (s) dBs.

We claim that

〈M, M〉t =∫ t

0| f (s)|2 ds.

We again verify this for a step process f . As before, we assume that s < tare among the points tj. We have

M2t −M2

s = ∑ fi−1 f j−1(Bti − Bti−1)(Btj − Btj−1),

where the sum is over those i and j such that ti ≤ t, tj ≤ t and at leastone of ti−1 and tj−1 is greater or equal o s. For an off-diagonal term, say,i < j, the term vanishes after conditioning on Ftj−1 , hence also vanishesafter conditioning on Fs. For a diagonal term i = j, since tj−1 ≥ s, by firstconditioning on Ftj−1 and then conditioning on Fs we have

E[| f j−1|2(Btj − Btj−1)

2|Fs

]= E

[| f j−1|2(tj − tj−1)|Fs

].

It follows that

E[M2

t −M2s |Fs

]= E

[∫ t

s| f (s)|2 ds

∣∣∣∣Fs

].

This shows that

M2t −

∫ t

0| f (s)|2 ds

is a martingale.

2. STOCHASTIC INTEGRALS WITH RESPECT TO BROWNIAN MOTION 47

2. Stochastic integrals with respect to Brownian motion

The general setting is as follow. We have a Brownian motion B withrespect to a filtration F∗ defined on a filtered probability space (Ω, F∗, P).We assume that the filtration F∗ satisfies the usual conditions.

DEFINITION 2.1. A real-valued function f : R+ ×Ω → R is called a stepprocess if there exists a nondecreasing sequence 0 = t0 < t1 < t2 < · · · increas-ing to infinity and a sequence of square-integrable random variables f j−1 ∈ Ftj−1

such thatft = f j−1, tj−1 ≤ t < tj.

The space of step processes is denoted by S . The space of step processes on [0, t] isdenoted by St.

Note that on an interval [tj−1, tj) where f is constant, it is measurablewith respect to the σ-algebra on the left endpoint of the interval, i.e., f =f j−1 ∈ Ftj−1 .

Let f be a step process as above. The stochastic integral of f with respectto Brownian motion B is the process

I( f )t =∫ t

0f (s) dBs =

∞

∑j=1

f j−1

(Btj∧t − Btj−1∧t

).

Of course this is a finite sum for each fixed t. If ti ≤ t < ti+1, then

I( f )t =i

∑j=1

f j−1(Btj − Btj−1) + fi(Bt − Bti).

It is easy to see that the definition of I( f )t is independent of the partitiontj

. As we have shown in the last section I( f ) is a continuous martingalewith the quadratic variation

〈I( f ), I( f )〉t =∫ t

0| f (s)|2ds.

In particular we have

E|I( f )t|2 = E

[∫ t

0| f (s)|2 ds

].

This relation allows us to extend the definition of stochastic integrals tomore general integrands by a limit procedure. Since the quadratic variationprocess of a Brownian motion corresponds to the Lebesgue measure on R+,a very wide class of processes can be used as integrand processes.

DEFINITION 2.2. A function f : R+ × Ω → R is called progressivelymeasurable with respect to the filtration F∗if for each fixed t the restriction f :[0, t]×Ω→ R is measurable with respect to the product σ-algebra B[0, t]×Ft.


EXAMPLE 2.3. Here are some examples of progressively measurableprocesses:

(1) A step process is progressively measurable.(2) A continuous and adapted process is progressively measurable.(3) Let τ be a finite stopping time. The the process

f (t, ω) = I[0,τ(ω)](s)

is progressively measurable. For a discrete τ, this can be verified directly.For a general τ, let

τn = ([2nτ] + 1)/2n.Then τn ↓ τ and I[0,τn(ω)](s)→ I[0,τ(ω)](s) for all (s, ω) ∈ R+ ×Ω.

For a progressively measurable process f , we define for each fixed T,

‖ f ‖22,T = E

[∫ T

0| f (s)|2ds

].

The progressive measurability assures that the integral on the right sidehas a meaning. We use L 2

T to denote the space of progressively measurableprocesses f on [0, T]×Ω with ‖ f ‖2,T < ∞. The norm ‖ · ‖2,T makes L 2

T intoa complete Hilbert space. We use L 2 to denote the space of progressivelymeasurable processes f such that ‖ f ‖2,T is finite for all T. It can be madeinto a metric space by introducing the distance function

d( f , g) =∞

∑n=1

‖ f − g‖2,n

1 + ‖ f − g‖2,n.

Note that ST ⊂ L 2 and S ⊂ L 2, and there is an obvious restriction mapL 2 → L 2

t .We now show how to define the stochastic integral

I( f )t =∫ t

0f (s) dBs

for every process in L 2. We need the following approximation result.

LEMMA 2.4. ST is dense in L 2T . In other words, for any square integrable

progressively measurable process f ∈ L 2T , there exists a sequence fn of step

processes such that ‖ fn − f ‖2,T → 0.

PROOF. First note that the set of uniformly bounded elements in L 2T

is dense. Suppose that f is a progressively measurable process uniformlybounded on [0, T]×Ω. Define

fh(t, ω) =1h

∫ t

t−hf (s, ω) ds.

Each fh is continuous and adapted. From real analysis we also know that

limh→0

∫ T

0| fh(t, ω)− f (t, ω)|2 ds = 0.


It follows that ‖ fh − f ‖2,T → 0 as h → 0. This shows that the set of uni-formly bounded continuous and adapted processes is dense in L 2

T .Finally for a continuous, adapted, and uniformly bounded process f on

[0, T] we define

fn(t) = f(

j− 1n

),

j− 1n≤ t <

jn

.

Each fn ∈ ST and fn(t, ω) → f (t, ω) for all (t, ω) ∈ [0, T]×Ω by continu-ity. Hence ‖ fn − f ‖2,T → 0 by the dominated convergence theorem. Thelemma is proved.

We are now ready to define the stochastic integral of a progressivelymeasurable and square integrable process with respect to a Brownian mo-tion. Suppose that f ∈ L 2. By restricting to [0, t] we can regard it as anelement in L 2

t for any t. Let fn be a sequence of step processes on [0, T]such that ‖ fn − f ‖2,T → 0. The stochastic integral I( fn)t is well defined for0 ≤ t ≤ T as a Riemannan sum and is a square integrable martingale. Wehave

I( fm)t − I( fn)t =∫ t

0[ fm(s)− fn(s)] dBs.

By Doob’s martingale inequality,

E

[max

0≤t≤T|I( fm)t − I( fn)t|2

]≤ 4E|I( fm − fn)T|2

= 4E

[∫ T

0| fm(s)− fn(s)|2ds

]= 4‖ fm − fn‖2

2,T.

This shows that I( fn)t is a Cauchy sequence in L2(Ω, P). By the complete-ness of L2(Ω, P), the limit I( f )t = limn→∞ I( fn)t exists and we define∫ t

0f (s) dBs = lim

n→∞

∫ t

0fn(s) dBs.

Since ‖ fn − f ‖2,T1 → 0 implies ‖ fn − f ‖2,T2 for T2 ≤ T1, it is easy to see thatI( f )t thus defined is independent of T and the choice of the approximatingsequence fn.

We have only defined I( f )t, t ≥ 0 as a collection of random variables.We can do more.

THEOREM 2.5. Suppose that f ∈ L 2 is a square integrable, progressivelymeasurable process. Then the stochastic integral

I( f )t =∫ t

0f (s) dBs

is a continuous martingale. Its quadratic variation process is

〈I( f ), I( f )〉t =∫ t

0f (s)2 ds.


PROOF. The martingale property of I( f ) is inherited from the sameproperty of I( fn) because this property is preserved when passing to thelimit in L2(Ω, F∗, P) as n → ∞. We show that it has continuous samplepaths. From

E

[max

0≤t≤T|I( fm)t − I( fn)t|2

]≤ 4‖ fm − fn‖2

2,T

we have by Chebyshev’s inequality,

P

[max

0≤t≤T|I( fm)t − I( fn)t| ≥ ε

]≤ 4

ε2 ‖ fm − fn‖22,t.

Since ‖ fm − fn‖2,T → 0 as m, n → ∞, by choosing a subsequence if neces-sary, we may assume that ‖ fn − fn+1‖2,T ≤ 1/n3, hence

P

max

0≤t≤T|I( fn)t − I( fn+1)t| ≥

1n2

≤ 4

n2 .

By the Borel–Cantelli lemma, there is a set Ω0 with P Ω0 = 1 with thefollowing property: for any ω ∈ Ω0, there is an n(ω) such that

max0≤t≤T

|I( fn)t(ω)− I( fn+1)t(ω)| ≤ 1n2

for all n ≥ n(ω). It follows that I( fn)t(ω) converges uniformly on [0, T].Since each I( fn)t(ω) is continuous in t, the limit, which necessarily coincidewith I( f )t(ω) with probability 1 for all t, must also be continuous. Wetherefore have shown that the stochastic integral I( f )t has a version withcontinuous sample paths.

We have shown before that for the step process fn the process

I( fn)2t −

∫ t

0fn(s)2 ds

is a martingale. For each fixed t, as n → ∞, the above random variableconverges in L1(Ω, P) to

I( f )2t −

∫ t

0f (s)2 ds.

Therefore the martingale property can be passed to the limiting process,which identifies the quadratic variation process of the stochastic integral asstated in the proposition.

COROLLARY 2.6. Suppose that f , g ∈ L 2. Then

〈I( f ), I(g)〉t =∫ t

0f (s)g(s) ds.

PROOF. This follows from the equality

〈M, N〉t =〈M + N, M + N〉t − 〈M− N, M− N〉t

4for two continuous martingales M and N.


If Z ∈ Fs, we have

Z∫ t

sfu dBu =

∫ t

sZ fu dBu.

This can be considered to be self-evident, even though it requires an ap-proximation argument to verify. The same remark can be said of the iden-tity

∫ t

sfu dBu =

∫ t

0fu dBu −

∫ s

0fu dBu.

Either we define the left side by the right side or we consider the left side asthe stochastic integral with respect to Bs+u, u ≥ 0, which is a Brownianmotion with respect to the filtration Fs+∗ = Fs+u, u ≥ 0, the end resultis the same. However, if we replace s or t by a random time, the meaningof the two and other similar equalities may become ambiguous, especiallywhen the random time is not a stopping time. In this respect, if we restrictourselves to stopping times, the following results may help us clarify thesituation.

PROPOSITION 2.7. Suppose that f ∈ L 2T and τ ≤ T a stopping time.

(1) We have

∫ τ

0fs dBs =

∫ T

0I[0,τ](s) fs dBs.

(2) If Z ∈ Fτ is a bounded bounded random variable, then

Z∫ T

τf dBs =

∫ T

0ZI[τ,∞)(s) fs dBs.

PROOF. We will verify the equalities for discrete stopping times andstep processes. In general we can approximate the stopping times σ and τby discrete stopping times from above and the integrand process f by stepprocesses in the ‖ · ‖2,T norm. The equalities are clearly preserved whenpassing to the limit under such approximations.

Let 0 = t0 < t1 < · · · < tn = T be a partition such that f = f j−1 ∈ Ftj−1

for tj−1 ≤ t < tj and τ takes values only in the sequence

tj

.


(1) Note that Is<τ is a step process. We have

I( f )τ =n

∑i=1

Iτ=ti

∫ ti

0fs dBs

=n

∑i=1

Iτ=tji

∑j=1

f j−1(Btj − Btj−1)

=n

∑j=1

f j−1(Btj − Btj−1)n

∑i=j

Iτ=ti

=n

∑j=1

Itj−1<τ f j−1(Btj − Btj−1)

=∫ T

0I[0,τ)(s) fs dBs.

(2) Note that ZIτ≤s is a step process and belongs to Fs for fixed s. Wehave

Z∫ T

σfs dBsZ

n

∑i=1

ZIτ=ti

∫ ti

0fsdBs

=n

∑i=1

Iτ=ti

n

∑j=i+1

Z f j−1(Btj − Btj−1)

=n

∑j=1

Z f j−1(Btj − Btj−1)j−1

∑i=1

Iτ=ti

=n

∑j=1

ZIτ≤tj−1 f j−1(Btj − Btj−1)

=∫ T

0ZI[τ,∞)(s) fs dBs.

EXAMPLE 2.8. Consider the stochastic integral∫ t

0Bs dBs = lim

n

∑i=1

Btj−1(Btj − Btj−1).

Using the identity 2a(b− a) = b2 − a2 + (b− a)2 we have

2n

∑i=1

Btj−1(Btj − Btj−1) = B2t +

n

∑i=1

(Btj − Btj−1)2.

The last sum converges to the quadratic variation 〈B, B〉t = t. Hence wehave

2∫ t

0Bs dBs = B2

t − t,

which is indeed a martingale.

3. EXTENSION TO MORE GENERAL INTEGRANDS 53

EXAMPLE 2.9. One may wonder what will happen if in the approximat-ing sum of a stochastic integral we do not take the value of the integrandat the left endpoint of a partition interval. For example, what is the sum

n

∑j=1

Btj(Btj − Btj−1)?

This is not hard to figure out. The difference between this sum and the usu-al sum taking the left endpoint value of the integrand is just the quadraticvariation along the partition

n

∑j=1

(Btj − Btj−1)2,

hence we ave

lim|∆|→0

n

∑j=1

Btj(Btj − Btj−1) =∫ t

0Bs dBs + t.

This example shows where to take the value of the integrand on each par-tition interval really matters.

3. Extension to more general integrands

We have defined stochastic integral with respect to a Brownian motionfor integrand process f ∈ L 2. These are progressively measurable process-es f such that

E

[∫ t

0f 2s ds

]< ∞

for all t. This integrability is too restrictive. Without too much effort we candefine a much wider class of integrand processes.

DEFINITION 3.1. We use L 2 to denote the space of progressively measurableprocesses f such that

P

[∫ t

0f 2s ds < ∞

]= 1

for all t.

This is a class of process wider than L 2 and is sufficient for our appli-cations. For example, all adapted continuous processes belong to L 2

loc. Wewill extend the definition of stochastic integrals to processes in L 2

loc. Theresulting stochastic integral process is no longer a square integrable mar-tingale but a local martingale.

Recall the definition of a local martingale.

DEFINITION 3.2. We say that M = Mt, t ≥ 0 is a local martingale ifthere exists a sequence of stopping times τn ↑ ∞ such that the stopped processesMτ = Mt∧τn , t ≥ 0 are martingales.


For continuous local martingale M with M0 = 0, we can always take

τn = inf t ≥ 0 : |Mt| ≥ n .

Suppose that f ∈ L 2loc. Define the stopping time

τn = inf

t > 0 :∫ t

0| f (s)|2ds ≥ n

.

the assumption that f ∈ L 2loc implies that τn ↑ ∞ (with probability one, if

one wants to be very precise). Now consider the process

fn(s) = f (s)Is≤τn.

It is clear that f ∈ L 2, in fact ‖ fn‖2,T ≤ n for all T. The stochastic integralI( fn) is well-defined and is a square–integrable continuous martingale. Wenow define the stochastic integral I( f )t as follows:

I( f )t = I( fn)t, t ≤ τn.

In order that this definition make sense, we have to verify the consistency:if m ≤ n, then

I( fn)t = I( fm)t, t ≤ τm.This is immediate. Indeed,

I( fn)t∧τm =∫ t

0fn Is≤τm dBs =

∫ t

0fmdBs = I( fm)t.

From definition we have

I( f )t∧τn = I( fn)t,

which shows that I( f ) is a continuous local martingale.

EXAMPLE 3.3. The process BeB2 6∈ L 2 because E[|Bt|2eB2

t

]is infinite

for t ≥ 1/2. However, since it is continuous it belongs to L 2loc and its

stochastic integral with respect to Brownian motion is well defined. In fact,Ito’s formula will show that

2∫ t

0BseB2

s dBs = eB2t −

∫ t

0

[1 + 2B2

s]

eB2s ds.

4. Stochastic integration with respect to continuous local martingales

We have discussed stochastic integration with respect to Brownian mo-tion so that only the martingale properties of Brownian motion is used. Forthis reason not much needs to be changed if we replace Brownian motionby a general continuous martingale. All we need to do is to replace thequadratic variation process of Brownian motion 〈B, B〉t = t by 〈M.M〉t,which is a continuous increasing process. More specifically, the theory ofstochastic integration with respect to Brownian motion is based on severalproperties of Brownian motion:

(1) B is a continuous square integrable martingale;(2) The quadratic variation process is 〈B, B〉t = t.

4. STOCHASTIC INTEGRATION WITH RESPECT TO CONTINUOUS LOCAL MARTINGALES55

(3) If f is a step process, then

E

[(∫ t

0f dBs

)2]= E

[∫ t

0| fs|2 ds

].

(4) The space of step processes ST is dense in the space of progressivelymeasurable and square integrable processes L 2

T .For a theory of stochastic integration with respect to a general continu-

ous martingale M, we make the following observations:(1) By the usual stopping time argument, we can restrict ourselves ini-

tially to uniformly bounded continuous martingales.(2) The quadratic variation process 〈M, M〉t is continuous, adapted, and

increasing. Therefore integration of a progressively measurable processwith respect to 〈M, M〉 is well defined.

(3) For a bounded step process f we have

E

∣∣∣∣∫ t

0fs dMs

∣∣∣∣2 = E

[∫ t

0f 2s d〈M, M〉s

].

(4) We can introduce the norm

‖ f ‖2,T;M =

√E

[∫ t

0f 2s d〈M, M〉s

].

There does not seem to be a good description of the closure of the spaceof ST under this norm that is good for all M, but at least all progressivelymeasurable and continuous processes with finite norm can be approximat-ed by step processes.

It is also possible develop a theory of stochastic integration with respectto a general, not necessarily continuous martingale. Since we will mostlyonly deal with continuous martingales, we will restrict ourselves to contin-uous local martingales.

Suppose that M is a continuous local martingale and f ∈ S is a stepprocess. The stochastic integral of f with respect to the martingale M is

I( f )t =∫ t

0f dMs = ∑

jf j−1

(Mtj−1∧t −Mtj∧t

).

Just as in the case of Brownian motion we have

E

∣∣∣∣∫ t

0fs dMs

∣∣∣∣2 = E

[∫ t

0| fs|2d〈M, M〉

].

Let L 2(M) be the space of progressively measurable processes f such that

‖ f ‖22,T;M = E

[∫ T

0| fs|2d〈M, M〉s

]< ∞

for all t ≥ 0. It can be shown that ST is dense in L 2(M)T (see Karatzasand Shreve [7]). Thus by the usual approximation argument, we see that


the stochastic integral

I( f )t =∫ t

0fs dMs

is well defined for all f ∈ L 2(M) and the quadratic variation process is

〈I( f ), I( f )〉t =∫ t

0| fs|2 d〈M, M〉s.

Although it is not easy to find a good description of L 2(M) that fits allcontinuous martingales, it does contain all continuous adapted processessuch that ‖ f ‖2,T;M is finite for all T.

By the standard stopping time argument we have used in the case ofBrownian motion, the stochastic integral

I( f )t =∫ t

0fs dMs

can be defined for a continuous local martingale M and a progressivelymeasurable process f such that

P

[∫ t

0| fs|2 d〈M, M〉s < ∞

]= 1

for all t. Under these conditions, the stochastic integral I( f ) is a continuouslocal martingale.

EXAMPLE 4.1. Let M be a uniformly bounded continuous martingale.As in the case of Brownian motion we have

M2t −M2

0 = 2n

∑i=1

Mtj−1(Mtj −Mtj−1) +n

∑i=1

(Mtj −Mtj−1)2.

The sum after the equal sign converges to the stochastic integral∫ t

0Ms dMs.

Therefore the second sum must also converge. It is clear that the limit of thesecond sum is an continuous, increasing, and adapted process. Hence bythe Doob-Meyer decomposition theorem, the limit must be the quadraticvariation process, i.e.,

lim|∆|→0

n

∑i=1

(Mtj −Mtj−1)2 = 〈M, M〉t

and

M2t = M2

0 + 2∫ t

0Ms dMs + 〈M, M〉t.

The convergence of the quadratic variation process takes place in L2(Ω, P).For a general local martingale, it can be shown by routine argument thatthe convergence takes place at least in probability.

5. ITO’S FORMULA 57

5. Ito’s formula

Ito’s formula is the fundamental theorem for stochastic calculus. Let usrecall that the fundamental theorem of calculus states that if F is a continu-ously differentiable function then

F(t)− F(0) =∫ t

0F′(s) ds.

Let 0 = t0 < t1 < · · · < tn = t be a partition of the interval [a, b]. We have

F(t)− F(0) =n

∑i=1

[F(tj)− F(tj−1)

].

Since F′ is continuous, by the mean value theorem, there is a point ξi ∈[tj−1, tj] such that

F(tj)− F(tj−1) = F(ξ j)(tj − tj−1).

Hence, by the definition of Riemann integrals we have as |∆| → 0,

F(b)− F(a) =n

∑j=1

F′(ξ j)(tj − tj−1)→∫ t

0F′(s) ds.

What happens to this proof if we replace t by Bt. In this case ξ j will be apoint between Btj−1 and Btj . We have to deal with the sum

n

∑i=1

F′(ξ j)(Btj − Btj−1).

Here the argument has to depart from what we have done above. As EX-AMPLE 2.9 shows, the place where we take the value of the integrand oneach partition interval can change the limit of the Riemann sum. Thereforewe should not expect that the above sum will converges to the stochastic

integral∫ t

0F′(Bs) dBs, for which the left endpoint value of the integrand is

used in each partition interval. The method to remedy this situation is totake one more term of the Taylor expansion

F(Btj)− F(Btj−1) = F′(Btj−1)(Btj − Btj−1) +12

F′′(ξ j)(Btj − Btj−1)2.

The sum of the first term on the right side will converge to the stochasticintegral as usual. The crucial observation is that because Brownian motionhas finite quadratic variation, the sum of the second term on the right side

n

∑i=1

F′′(ξ j)(Btj − Btj−1)2 →

∫ t

0F′′(Bs) ds

as |∆| → 0. The final result is the well know Ito’s formula

F(Bt) = F(B0) +∫ t

0F′(Bs) dBs +

12

∫ t

0F′′(Bs) ds.


On the right side, the first term is a continuous martingale, and the sec-ond term is a process of bounded variation. Thus if F is continuous anduniformly bounded together with its first and second derivatives, then thecomposition F(Bt) is a semimartingale and Ito’s formula gives an explicitexpression for its Doob-Meyer decomposition.

The proof of Ito’s formula for for a general semimartingale Z will notbe a lot more difficult than the case of Brownian motion. For this reason, inthis section we will prove it in this generality.

THEOREM 5.1. Suppose that F ∈ C2(R) and Z is a semimartingale. Then

(5.1) F(Zt) = F(Z0) +∫ t

0F′(Zs) dZs +

12

∫ t

0F′′(Zs) d〈Z, Z〉s.

Recall that a semimartingale has the form

Zt = Mt + At,

where M is a local martingale and A a process of bounded variation. Wefirst use a stopping time argument to reduce the proof to the case where Z0,M, 〈M, M〉, and A are all uniformly bounded and F together with its firstand second derivatives are uniformly bounded and uniformly continuous.First of all, let ΩN = |Z0| ≤ N and define

PN(C) =P(C ∩ΩN)

P(Ω0).

It is clear that under the PN , the process Z is still a semimartingale withthe same decomposition Z = M + A and PN |Z0| ≤ N = 1, i.e., Z0 isbounded. Next, define the stopping time

τN = inf t : |Mt −M0|+ 〈M, M〉t + |A|t ≥ N .

The stopped processes ZτN = MτN + AτN and 〈MτN , MτN 〉t = 〈M, M〉τNt

have the bounded properties we desired. Define a function FN ∈ C2(R)such that it coincides with F on [−2N, 2N] and vanishes outside [−3N, 3N].On the probability space (ΩN , F∗ ∩ ΩN , PN) suppose that we have theformula

FN(ZτN ) = FN(ZτN ) +∫ t

0F′N(ZτN

s ) dZs +12

∫ t

0F′′N(ZτN

s ) d〈ZτN , ZτN 〉s.

Replacing t there by t ∧ τN we see that

F(Zt∧τN ) = F(Z0) +∫ t∧τN

0F′(Zs) dZs +

12

∫ t∧τN

0F′′(Zs) d〈Z, Z〉s.

Here we have used the definition of stochastic integral with respect to alocal martingale. This shows that the formula (5.1) holds on ΩN and t < τN .Finally from P ΩN ↑ 1 and P τN ↑ ∞ = 1, we see that Ito’s formulaholds without any extra conditions.

5. ITO’S FORMULA 59

After these reductions, we start the proof of Ito’s formula itself. ByTaylor’s formula with remainder we have

F(Ztj)− F(Ztj−1) = F′(Ztj−1)(Ztj − Ztj−1) +12

F′′(ξi)(Ztj − Ztj−1)2,

where ξ is a point between Ztj−1 and Ztj . For simplicity let us denote

∆Zj = Ztj − Ztj−1 , ∆Mj = Mtj −Mtj−1 , ∆〈M〉j = 〈M〉tj − 〈M〉tj−1 .

With these notations we can write

F(Zt)− F(Z0) =n

∑i=1

F′(Ztj−1)∆Zj +12

n

∑i=1

F′′(ξi)(∆Zj)2.

The first sum converges to the stochastic integral in L2(Ω, P):n

∑i=1

F′(Ztj−1)∆Zj →∫ t

0F′(Zs) dZs.

The real work starts with the proof of

(5.2)n

∑i=1

F′′(ξi)(∆Zj)2 →

∫ t

0F′′(Zs) d〈Z, Z〉s.

We will prove this through a series of three replacements

F′′(ξi)(∆Zj)2 ⇒ F′′(ξi)(∆Mj)

2 ⇒ F′′(Ztj−1)(∆Mj)2 ⇒ F′′(Ztj−1)∆〈M〉j.

We will show that each replacement produces an error which will vanish inprobability as the mesh of the partition ∆| → 0. After these replacementswe will have

n

∑i=1

F′′(Ztj−1)∆〈M〉j →∫ t

0F′′(Zs) d〈M, M〉s,

which will complete the proof.(1) From (∆Zj)

2 − (∆Mj)2 = ∆Aj(∆Zj + ∆Mj) we have∣∣∣∣∣ n

∑i=1

F′′(ξi)(∆Zj)

2 − (∆Mj)2∣∣∣∣∣ ≤ ‖F′′‖∞|A|t max

1≤i≤n|∆Zj + ∆Mj|.

By sample path continuity we have

lim|∆|→0

max1≤i≤n

|∆Zj + ∆Mj| = 0

almost surely. Hence the error produced by the first replacement vanishesin probability as |∆| → 0.

(2) We have∣∣∣∣∣ n

∑i=1

[F′′(ξi)− F′′(Ztj−1)

](∆Mj)

2

∣∣∣∣∣ ≤ max1≤i≤n

|F′′(ξi)− F′′(Ztj−1)|n

∑i=1

(∆Mj)2.


Again by sample path continuity we have

max1≤i≤n

|F′′(ξi)− F′′(Ztj−1)| → 0

almost surely. On the other hand, by EXAMPLE 4.1,

lim|∆|→0

n

∑i=1

(∆Mj)2 = 〈M, M〉t

in L2(Ω, P). Therefore the error produced by this replacement vanishes inprobability as |∆| → 0.

(3) This is the most delicate replacement of the three. The error is

En =n

∑i=1

F′′(Ztj−1)(∆Mj)

2 − ∆〈M〉j

.

The square |En|2 is a sum of n2 terms. From EXAMPLE 4.1

(∆Mj)2 − ∆〈M〉j = 2

∫ tj

tj−1

(Ms −Mtj−1) dMs.

This is a stochastic integral with respect to a martingale, hence the condi-tional expectation of this expression with respect to Ftj−1 is clearly zero. Foran off-diagonal term, say i < j, in |En|2, its conditional expectation with re-spect to Ftj−1 is therefore zero. Hence all off-diagonal term has expectationvalue zero. For the diagonal term we have

E

[(∫ tj

tj−1

(Ms −Mtj−1) dMs

)2 ∣∣∣∣Ftj−1

]= E

[∫ tj

tj−1

(Ms −Mtj−1)2 d〈M, M〉s

].

Therefore the expected value of this off-diagonal term is bounded by

‖F′′‖2∞ E

[∆〈M〉j max

tj−1≤s≤tj|Ms −Mtj−1 |

].

Adding the diagonal terms together we obtain

E|En|2 ≤ ‖F′′‖2∞ E

[〈M〉t max

1≤j≤nmax

tj−1≤s≤tj|Ms −Mtj−1 |

].

Note that under our assumption, the expression under the expectation isuniformly bounded. This expectation converges to zero by sample pathcontinuity and the dominated convergence theorem.

With this final replacement, we have completed the proof of (5.2) andalso the proof of Ito’s formula.

We end this section with the statement of Ito’s formula for multidimen-sional semimartingales. The basic ingredients of its proof have alreadyexplained in great detail in the proof of the one dimensional case. Sincenothing will be learned from its tedious proof, we will justifiably omit it.

6. DIFFERENTIAL NOTATION AND STRATONOVICH INTEGRALS 61

THEOREM 5.2. Suppose that Z = (Z1, . . . , Zn) be an Rn valued semimartin-gale. Then for any F ∈ C2(Rn).

F(Zt) = F(Z0) +n

∑i=1

∫ t

0Fxi(Zs) dZi

s +12

n

∑i,j=1

∫ t

0Fxixj(Zs) d〈Zi, Zj〉s.

6. Differential notation and Stratonovich integrals

This section does not contain new results. We introduce differentialnotation for stochastic calculus and Stratonovich integrals, a form of sto-chastic integral more restrictive than Ito integrals.

We often work with the following classes of processes:M = the space of continuous local martingales;A = the space of continuous processes of bounded variation;I = the space of increasing processes;Q= the space of semimartingalesH = the space of integrand processes.

Every process of bounded variation is the difference of two increasing pro-cesses: A = I −I . A semimartingale is the sum of a local continuousmartingale and a process of bounded variation: Q = M +A

It is often convenient to write equalities among these processes in a d-ifferential form. We have defined the meaning of an expression such as∫ t

0HsdXs for H ∈ H and X ∈ Q. What exactly is the meaning a differ-

ential expression such as dX? We should consider dX as a symbol for theequivalence class of semimartingales Y such that

Xt − Xs = Yt −Ys

for all s ≤ t. The equivalent classes of semimartingale differentials is de-noted by dQ. We can carry out some purely symbolic calculations. To startwith we can set ∫ t

sdXu = Xt − Xs.

We can define a multiplication in Q by setting

dX · dY = d〈X, Y〉.

We also define the multiplication of an element from Q by an element from

H : HdX is the equivalence class containing the semimartingale∫ t

0Hs dXt

s.

Here are some properties of these operations:(1) dQ · dQ ⊂ dA ;(2) dQ · dA = 0;(3) H(dX · dY) = (H · dX) · dY;(4) H1(H2 · dX) = (H1H2)dX.


With these formal symbolic notations, the multi-dimensional Ito’s for-mula can be written as

d f (Zt) =n

∑j=1

fxi(Zt) dZit +

12

n

∑i,j=1

fxixj(Zt) dXi · dX j.

If we are willing to adopt Einstein’s summation convention, we may evendrop the two summation signs.

Suppose that X and Y are two semimartingales. We define a new semi-martingale differential by

X dY = X · dY +12

dX · dY.

This means that ∫ t

0Xs dYs =

∫ t

0Xs dYs +

12〈X, Y〉t.

This is called the Stratonovich integral. Unlike Ito integral, the Stratonovichintegral requires that the integrand process is also a continuous semimartin-gale, for in its definition we need the quadratic covariation of X and Y. Inthis sense Stratonovich integral is a weaker form of stochastic integral andits use is much more limited than Ito integrals. The main advantage of S-tratonovich integrals is that Ito’s formula takes a form similar to that of thefundamental theorem of calculus.

PROPOSITION 6.1. Let F ∈ C3(Rn) and Z = (Z1, . . . , Zn) an n-dimensionalsemimartingale. Then

d f (Zt) =n

∑i=1

Fxi(Zt) dZit.

PROOF. The right side is equal to

(6.1) Fxi(Zt) · dZjt +

12

Fxixj(Zt) · dZjt .

Using It&o’s formula we have

dFxi(Zt) = Fxixj(Zt) · dZit +

12

Fxixjxk(Zt) · dZj · dZk.

Thus the last term in (6.1) becomes12

Fxixj(Zt) · dZit · dZj

t +14

Fxixjxk(Zt) · dZit · dZj

t · dZkt .

The last triple sum is equal to zero because dZit · dZj

t · dZkt = 0. Therefore

the equality we wanted to prove is just Ito’s formula.

Finally we mention that Stratonovich integral can also be approximatedby a Riemann sums. It is instructive to compare this Riemann sum withthe one that approximates the corresponding Ito integral. The difference issubtle but crucial in making Ito integration such a successful theory.

7. THIRD ASSIGNMENT 63

THEOREM 6.2. Let X and Y be continuous semimartingales. Then∫ t

0Xs dYs = lim

|∆|→0

n

∑j=1

Xtj−1 + Xtj

2

[Ytj −Ytj−1

],

where the convergence is in probability.

PROOF. Rewrite the summation asn

∑j=1

Xtj−1

[Ytj −Ytj−1

]+

12

[Xtj − Xtj−1

] [Ytj −Ytj−1

].

The limit of the first term is the Ito integral∫ t

0XsdYs, that of the second

term is exactly the quadratic covariation〈X, Y〉t/2.

7. Third Assignment

EXERCISE 3.1. Let f be a (nonrandom) function of bound variation.Show that ∫ t

0fs dBs = ftBt −

∫ t

0Bs d fs,

where the last integral is understood to be a Lebesgue-Stieljes integral.

EXERCISE 3.2. Let F be an entire function in the complex plane C andZ = X + iY be the complex Brownian motion (meaning that X and Y areindependent standard Brownian motion). We have∫ t

0F′(Zs) dZs =

∫ t

0F′(Zs) dZs = F(Zt)− F(Z0).

EXERCISE 3.3. The price St of a stock with average rate of return µ andvolatility σ is usually described by the stochastic differential equation

dSt = St (µ dt + σ dBt) .

Using Ito’s formula to show that the solution of the above stochastic differ-ential equation is

St = S0 exp[

σBt +

(µ− σ2

2

)t]

.

EXERCISE 3.4. Let B = (B1, B2, B3) be a 3-dimensional Brownian mo-tion which does not start from zero. Using Ito’s formula to show that 1/|Bt|is a local martingale.

EXERCISE 3.5. Let B be an n-dimensional Brownian motion startingfrom zero and

Xt = |Bt| =√

n

∑i=1|Bi

t|2


be the radial process. Show that X satisfies the following Ito type stochasticdifferential equation:

Xt = Wt +n− 1

2

∫ t

0

dsXs

,

where W is a one-dimensional Brownian motion.

EXERCISE 3.6. Let M and N be two continuous local martingales. Showthat

|〈M, N〉t| ≤√〈M〉t〈N〉t.

EXERCISE 3.7. Suppose that X, Y and Z be three semimartingales. Arethe following relations true?

(1) X(Y dZ) = (XY) dZ;(2) X (Y Z) = (XY) dZ.

EXERCISE 3.8. A Brownian bridge is defined by

dXt = dBt −Xt dt1− t

, X0 = 0.

Show that X is a Gaussian process with mean zero and

E [XsXt] = min s, t − st.

Show that Xt, 0 ≤ t ≤ 1 and the reversed process X1−t, 0 ≤ t ≤ 1 havethe same law.

EXERCISE 3.9. Suppose that M and N are two bounded continuousmartingales which are independent. Show that MN is a continuous mar-tingale with respect to the filtration F M,N

∗ = σ Ms, Ns; s ≤ t generatedby M and N.

EXERCISE 3.10. Suppose that M is a strictly positive local martingale.Show that there is a local martingale N such that

Mt = M0 exp[

Nt −12〈N, N〉t

].

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Stochastic Integration and Ito’s Formulastaff.ustc.edu.cn/~wangran/Course/Hsu/Chapter 3... · Stochastic Integration and Ito’s Formula In this chapter we discuss Ito’s theory