Lecture 1: Brownian motion, martingales and Markov processessong/Gene Golub Summer School...Lecture 1: Brownian motion, martingales and Markov processes David Nualart Department of

Lecture 1: Brownian motion, martingales andMarkov processes

David Nualart

Department of MathematicsKansas University

Gene Golub SIAM Summer School 2016Drexel University

David Nualart (Kansas University) July 2016 1 / 54

Outline

1 Stochastic proceses. Brownian motion. Markov processes.2 Stopping times. Martingales.3 Stochastic integrals.4 Ito’s formula and applications.5 Stochastic differential equations.6 Introduction to Malliavin calculus.


Multivariate normal distribution

A random vector X = (X1, . . . ,Xn

) has the multivariate normal

distribution N(µ,⌃), if its characteristic function is

E

⇣

e

ihu,Xi⌘

= exp✓

ihu, µi � 12

u

T⌃u

◆

, u 2 Rn,

where µ 2 Rn and ⌃ is an n ⇥ n symmetric and nonnegative definitematrix.

µ = (E(X1), . . . ,E(Xn

))

⌃ij

= Cov(Xi

,Xj

)

If X has the N(µ,⌃) distribution, then Y = AX + b, where A is an m ⇥ n

matrix and b 2 Rm, has the N(Aµ+ b,A⌃A

T ) distribution.


If ⌃ is nonsingular, then X has a density given by

f (x) = (2⇡)�n

2 (det⌃)�12 exp

✓

�12(x � µ)T⌃�1(x � µ)

◆

.


Stochastic processes

A stochastic process X = {X

t

, t � 0} is a family of random variables

X

t

: ⌦ ! R

defined on a probability space (⌦,F ,P).

The probabilities on Rn, n � 1,

P

t1,...,tn = P � (Xt1 , . . . ,Xt

n

)�1

where 0 t1 < · · · < t

n

, are called the finite-dimensional marginal

distributions of the process X .

For every ! 2 ⌦, the mapping

t ! X

t

(!)

is called a trajectory of the process X .




t


X

t

: ⌦ ! R



P

t1,...,tn = P � (Xt1 , . . . ,Xt

n

)�1

where 0 t1 < · · · < t

n




t ! X

t

(!)





t


X

t

: ⌦ ! R



P

t1,...,tn = P � (Xt1 , . . . ,Xt

n

)�1

where 0 t1 < · · · < t

n




t ! X

t

(!)



Theorem (Kolmogorov’s extension theorem)Consider a family of probability measures

{P

t1,...,tn , 0 t1 < · · · < t

n

, n � 1}

such that :

(i) P

t1,...,tn is a probability on Rn

.

(ii) (Consistence condition) : If {t

k1 < · · · < t

k

m

} ⇢ {t1 < · · · < t

n

}, then

P

t

k1 ,...,tkm

is the marginal of P

t1,...,tn , corresponding to the indexes

k1, . . . , km

.

Then, there exists a stochastic process {X

t

, t � 0} defined in some

probability space (⌦,F ,P), which has the family {P

t1,...,tn} as

finite-dimensional marginal distributions.

Take ⌦ as the set of all functions ! : [0,1) ! R, F the �-algebragenerated by cylindrical sets, extend the probability from cylindrical setsto F , and set X

t

(!) = !(t).


Gaussian processes

X = {X

t

, t � 0} is called Gaussian if all its finite-dimensional marginaldistributions are multivariate normal.

The law of a Gaussian process is determined by the mean functionE(X

t

) and the covariance function

Cov(Xt

,Xs

) = E((Xt

� E(Xt

))(Xs

� E(Xs

))).

Suppose µ : R+ ! R, and � : R+ ⇥ R+ ! R is symmetric andnonnegative definite :

n

X

i,j=1

�(ti

, tj

)ai

a

j

� 0, 8 t

i

� 0, a

i

2 R.

Then there exists a Gaussian process with mean µ and covariancefunction �.


Equivalent processes

Two processes, X , Y are equivalent (or X is a version of Y ) if for allt � 0,

P{X

t

= Y

t

} = 1.

Two equivalent processes may have quite different trajectories. Forexample, the processes X

t

= 0 for all t � 0 and

Y

t

=

⇢

0 if ⇠ 6= t

1 if ⇠ = t

where ⇠ � 0 is a continuous random variable, are equivalent, becauseP(⇠ = t) = 0, but their trajectories are different.Two processes X and Y are said to be indistinguishable if

X

t

(!) = Y

t

(!)

for all t � 0 and for all ! 2 ⌦⇤, with P(⌦⇤) = 1.

Exercise : Two equivalent processes with right-continuous trajectories areindistinguishable.




P{X

t

= Y

t

} = 1.


t


Y

t

=

⇢

0 if ⇠ 6= t

1 if ⇠ = t

where ⇠ � 0 is a continuous random variable, are equivalent, becauseP(⇠ = t) = 0, but their trajectories are different.

Two processes X and Y are said to be indistinguishable if

X

t

(!) = Y

t

(!)






P{X

t

= Y

t

} = 1.


t


Y

t

=

⇢

0 if ⇠ 6= t

1 if ⇠ = t

where ⇠ � 0 is a continuous random variable, are equivalent, becauseP(⇠ = t) = 0, but their trajectories are different.Two processes X and Y are said to be indistinguishable if

X

t

(!) = Y

t

(!)




Regularity of trajectories

Theorem (Kolmogorov’s continuity theorem)Suppose that X = {X

t

, t 2 [0,T ]} satisfies

E(|Xt

� X

s

|�) K |t � s|1+↵,

for all s, t 2 [0,T ], and for some constants �,↵ > 0. Then, there exists a

version

e

X of X such that, if � < ↵/�,

|eXt

� e

X

s

| G� |t � s|�

for all s, t 2 [0,T ], where G� is a random variable.

The trajectories of eX are Holder continuous of order � for any � < ↵/�.


Sketch of the proof :

(i) Suppose T = 1. Take � < ↵/� and set Dn

= { k

2n

, 0 k 2n} andD = [

n�1Dn

. From Chebychev’s inequality,

P( max1k2n

|Xk

2n

� X

k�12n

| � 2��n) 2n

X

k=1

P(|Xk

2n

� X

k�12n

| � 2��n)

2n

X

k=1

2��n

E [|Xk

2n

� X

k�12n

|�)

K 2�n(↵��).

Because this series of probabilities is convergent, from the Borel-Cantellilemma, there is a set ⌦⇤ 2 F with P(⌦⇤) = 1 such that for all ! 2 ⌦⇤,there exists N(!) with

|Xk

2n

(!)� X

k�12n

(!)| < 2��n, 8n � N(!), 81 k 2n.


(ii) Suppose that s, t 2 D are such that

|s � t | 2�n, n � N.

Then , there exists two increasing sequences s

k

2 Dk

and t

k

2 Dk

,k � n, converging to s and t respectively, and such that|s

k+1 � s

k

| 2�(k+1), |tk+1 � t

k

| 2�(k+1) and |sn

� t

n

| 2�n. Then,from the decomposition

X

s

� X

t

=1X

i=n

(Xs

i+1 � X

s

i

) + (Xs

n

� X

t

n

) +1X

i=n

(Xt

i

� X

t

i+1)

we obtain|X

t

� X

s

| 21 � 2��

2��n.

This implies that the paths t ! X

t

(!) are �-Holder on D for all ! 2 ⌦⇤,which allows us to conclude the proof. ⇤


Brownian motion

A stochastic process B = {B

t

, t � 0} is called a Brownian motion if :i) B0 = 0 almost surely.ii) Independent increments : For all 0 t1 < · · · < t

n

the incrementsB

t

n

� B

t

n�1 , . . . ,Bt2 � B

t1 , are independent random variables.iii) If 0 s < t , the increment B

t

� B

s

has the normal distributionN(0, t � s).

iv) With probability one, t ! B

t

(!) is continuous.


PropositionProperties i), ii), iii) are equivalent to :

(?) B is a Gaussian process with mean zero and covariance

�(s, t) = min(s, t).

Proof :

a) Suppose i), i) and iii). The distribution of (Bt1 , . . . ,Bt

n

), for0 < t1 < · · · < t

n

, is normal, because this vector is a lineartransformation of

�

B

t1 ,Bt2 � B

t1 , . . . ,Bt

n

� B

t

n�1

�

which has independentand normal components.The mean is zero, and for s < t , the covariance is

E(Bs

B

t

) = E(Bs

(Bt

� B

s

+ B

s

)) = E(Bs

(Bt

� B

s

)) + E(B2s

) = s.

b) The converse is also easy to show. ⇤


PropositionProperties i), ii), iii) are equivalent to :

(?) B is a Gaussian process with mean zero and covariance

�(s, t) = min(s, t).

Proof :

a) Suppose i), i) and iii). The distribution of (Bt1 , . . . ,Bt

n

), for0 < t1 < · · · < t

n

, is normal, because this vector is a lineartransformation of

�

B

t1 ,Bt2 � B

t1 , . . . ,Bt

n

� B

t

n�1

�

which has independentand normal components.The mean is zero, and for s < t , the covariance is

E(Bs

B

t

) = E(Bs

(Bt

� B

s

+ B

s

)) = E(Bs

(Bt

� B

s

)) + E(B2s

) = s.

b) The converse is also easy to show. ⇤


First construction of the Brownian motion

1. The function �(s, t) = min(s, t) is symmetric and nonnegative definitebecause it can be written as

min(s, t) =Z 1

01[0,s](r)1[0,t](r)dr ,

son

X

i,j=1

a

i

a

j

min(ti

, tj

) =n

X

i,j=1

a

i

a

j

Z 1

01[0,t

i

](r)1[0,tj

](r)dr

=

Z 1

0

"

n

X

i=1

a

i

1[0,ti

](r)

#2

dr � 0.

Therefore, by Kolmogorov’s extension theorem there exists a Gaussianprocess B with zero mean and covariance function min(s, t).


2. The process B satisfies

E

h

(Bt

� B

s

)2k

i

=(2k)!

2k

k !(t � s)k , s t

for any k � 1, because the distribution of B

t

� B

s

is N(0, t � s).

3. Therefore, by the Kolmogorov’s continuity theorem, there exist a versione

B of B, such that eB has Holder continuous trajectories of order � for any� < k�1

2k

on any interval [0,T ]. This implies that the paths are �-Holderon [0,T ] for any � < 1

2 and for any T > 0.


Second construction of Brownian motionFix T > 0.

(i) {e

n

, n � 0} is an orthonormal basis of L

2([0,T ]).(ii) {Z

n

, n � 0} are independent N(0, 1) random variables.Then, as N ! 1,

sup0tT

�

�

�

�

�

N

X

n=0

Z

n

Z

t

0e

n

(s)ds � B

t

�

�

�

�

�

a.s.,L2

�! 0.

Notice that

E

"

N

X

n=0

Z

n

Z

t

0e

n

(r)dr

!

N

X

n=0

Z

n

Z

s

0e

n

(r)dr

!#

=N

X

n=0

Z

t

0e

n

(r)dr

!

✓

Z

s

0e

n

(r)dr

◆

=N

X

n=0

⌦

1[0,t], en

↵

L

2([0,T ])

⌦

1[0,s], en

↵

L

2([0,T ])

N!1!⌦

1[0,t], 1[0,s]↵

L

2([0,T ])= s ^ t .


In particular, if T = 2⇡, e0(t) =1p2⇡

and e

n

(t) = 1p⇡

cos(nt/2), for n � 1,we obtain the Paley-Wiener representation of Brownian motion :

B

t

= Z0tp2⇡

+2p⇡

1X

n=1

Z

n

sin(nt/2)n

, t 2 [0, 2⇡].

In order to use this formula to get a simulation of Brownian motion, wehave to choose some number M of trigonometric functions and anumber N of discretization points.


Third construction of Brownian motion

Let {⇠k

, 1 k n} be independent and identically distributed randomvariables with zero mean and variance one.

Define S

n

(0) = 0,

S

n

(kT

n

) =p

T

⇠1 + · · ·+ ⇠kp

n

, k = 1, . . . , n

and extend S

n

(t) to t 2 [0,T ] by linear interpolation.

Donsker Invariance Principle : The law of the random walk S

n

onC([0,T ]) converges to the Wiener measure, which is the law of theBrownian motion. That is, that for any continuous and bounded function' : C([0,T ]) ! R,

E('(Sn

))n!1�! E('(B)),


Simulations of Brownian motion


Basic properties

1. Selfsimilarity :

For any a > 0, the process {a

� 12B

at

, t � 0} is also a Brownian motion.


2. For any h > 0, the process {B

t+h

� B

h

, t � 0} is a Brownian motion.

3. The process {�B

t

, t � 0} is a Brownian motion.

4. Almost surely limt!1

B

t

t

= 0 and the process

X

t

=

(

tB1/t

, t > 00, t = 0

is a Brownian motion.

5. P(sups,t2[0,1]

|Bt

�B

s

|p|t�s|

= +1) = 1.

6. P(supt�0 B

t

= +1, inft�0 B

t

= �1) = 1.

7. Almost surely the paths of B are not differentiable at any point t � 0.


Quadratic variation

Fix a time interval [0, t ] and consider a partition

⇡ = {0 = t0 < t1 < · · · < t

n

= t}.

Define �t

k

= t

k

� t

k�1, �B

k

= B

t

k

� B

t

k�1 and |⇡| = max1kn

�t

k

.

PropositionThe following convergence holds in L

2:

lim|⇡|!0

n

X

k=1

(�B

k

)2 = t .

We can say that (�B

t

)2 ⇠ �t


Proof : Set ⇠k

= (�B

k

)2 ��t

k

. The random variables ⇠k

are independent andcentered. Thus,

E

2

4

n

X

k=1

(�B

k

)2 � t

!23

5 = E

2

4

n

X

k=1

⇠k

!23

5 =n

X

k=1

E

⇥

⇠2k

⇤

=n

X

k=1

h

3 (�t

k

)2 � 2 (�t

k

)2 + (�t

k

)2i

= 2n

X

k=1

(�t

k

)2 2t |⇡| |⇡|!0�! 0.

⇤Exercise : Using the Borel-Cantelli lemma, show that if {⇡n} is a sequence ofpartitions of [0, t ] such that

P

n

|⇡n| < 1, thenP

n

k=1 (�B

k

)2 convergesalmost surely to t .


Infinite total variation

Define

V

t

= sup⇡

n

X

k=1

|�B

k

|

Then,P(V

t

= 1) = 1.

In fact, using the continuity of the trajectories of the Brownian motion, wehave, on the set V < 1,

n

X

k=1

(�B

k

)2 supk

|�B

k

|

n

X

k=1

|�B

k

|!

V supk

|�B

k

| |⇡|!0�! 0.

Then, V < 1 contradicts the fact thatP

n

k=1 (�B

k

)2 converges in L

2 to t

as |⇡| ! 0.


Fine properties of the trajectories

L

´

evy’s modulus of continuity :

lim sup�#0

sups,t2[0,1],|t�s|<�

|Bt

� B

s

|p

2|t � s| log |t � s|= 1, a.s.

In contrast, the behavior at at single point is given by the law of iterated

logarithm :

lim supt#s

|Bt

� B

s

|p

2|t � s| log log |t � s|= 1, a.s.

for any s � 0.


Conditional expectation

Let X be an integrable random variable on a probability space (⌦,F ,P) andG ⇢ F a �-algebra.

DefinitionThe conditional expectation E(X |G) is a random variable Y safisfying :

(i) Y is G-measurable.

(ii) For all A 2 G,Z

A

XdP =

Z

A

YdP.

if X � 0, E(X |G) is the density of the measure µ(A) =R

A

XdP, restrictedto G, with respect to P.

By the Radon-Nikodym theorem, E(X |G) exists and it is unique almostsurely.


Properties of the conditional expectation

1. Linearity :

E(aX + bY |G) = aE(X |G) + bE(Y |G).

2. E (E(X |G)) = E(X ).

3. If X and G are independent, then E(X |G) = E(X ).

4. If X is G-measurable, then E(X |G) = X .

5. If Y is bounded and G-measurable, then

E(YX |G) = YE(X |G).

6. Given two �-fields B ⇢ G, then

E (E(X |B)|G) = E (E(X |G)|B) = E(X |B).


7. Let X and Z be such that :

(i) Z is G-measurable.(ii) X is independent of G.

Suppose that E (|h(X ,Z )|) < 1. Then,

E (h(X ,Z )|G) = E (h(X , z)) |z=Z

.


Markov processes

A filtration {Ft

⇢ F , t � 0} is an increasing family of �-fields.

A process {X

t

, t � 0} is Ft

-adapted if X

t

is Ft

-measurable for all t � 0.

DefinitionAn adapted process X

t

is a Markov process with respect to Ft

if for any s � 0,t > 0 and any f 2 C

b

(R),

E [f (Xs+t

)|Fs

] = E [f (Xs+t

)|Xs

], a.s.

This implies that X

t

is also an FX

t

-Markov process, whereFX

t

= �{X

u

, 0 u t}.

The finite-dimensional marginal distributions of a Markov process arecharacterized by the transition probabilities

p(s, x , s + t ,B) = P(Xs+t

2 B|Xs

= x).


Markov property of Brownian motion

TheoremThe Brownian motion B

t

is an FB

t

-Markov process such that, for any

f 2 C

b

(R), s � 0 and t > 0,

E [f (Bs+t

)|FB

s

] = (Pt

f )(Bs

),

where (Pt

f )(x) =R

R f (y) 1p2⇡t

e

� |x�y|22t

dy.

{P

t

, t � 0} is the the semigroup of operators associated with theBrownian motion :

P

t

� P

s

= P

t+s

P0 = Id


Proof :

We haveE [f (B

s+t

)|FB

s

] = E [f (Bs+t

� B

s

+ B

s

)|FB

s

].

Since B

s+t

� B

s

is independent of FB

s

, we obtain

E [f (Bs+t

)|FB

s

] = E [f (Bs+t

� B

s

+ x)]|x=B

s

=

Z

Rf (y + B

s

)1p2⇡t

e

� |y|22t

dy

=

Z

Rf (y)

1p2⇡t

e

� |Bs

�y|22t

dy = (Pt

f )(Bs

).

⇤


Multidimensional Brownian motion

B

t

= (B1t

, . . . ,Bd

t

) is called a d-dimensional Brownian motion if itscomponents are independent Brownian motions.

It is a Markov process with semigroup

(Pt

f )(x) =

Z

Rd

f (y)(2⇡t)�d

2 exp✓

� |x � y |2

2t

◆

.

The transition density p

t

(x , y) = (2⇡t)�d

2 exp⇣

� |x�y|22t

⌘

satisfies the heatequation

@p

@t

=12�p, t > 0,

with initial condition p0(x , y) = �x

(y).


Stopping times

Consider a filtration {Ft

, t � 0} in a probability space (⌦,F ,P), thatsatisfies the following conditions :

(i) If A 2 F is such that P(A) = 0, then A 2 F0.(ii) The filtration is right-continuous, that is, for every t � 0,

Ft

= \n�1F

t+ 1n

.

DefinitionA random variable T : ⌦ ! [0,1] is a stopping time with respect to a filtration{F

t

, t � 0} if{T t} 2 F

t

, 8t � 0.


Properties of stopping times

1. T is a stopping time if and only if {T < t} 2 Ft

for all t � 0.Proof :

{T < t} = [n

{T t � 1n

} 2 Ft

.

Conversely,

{T t} = \n

{T < t +1n

} 2 \Ft+ 1

n

= Ft

. ⇤

2. S _ T and S ^ T are stopping times.

3. Given a stopping time T ,

FT

= {A : A \ {T t} 2 Ft

, for all t � 0}.

is a �-field.

4. S T ) FS

⇢ FT

.


5. Let {X

t

, t � 0} be a continuous and adapted process. The hitting time ofa set A ⇢ R is defined by

T

A

= inf{t � 0 : X

t

2 A}.

Then, if A is open or closed, T

A

is a stopping time.

6. Let X

t

be an adapted stochastic process with right-continuous paths andT < 1 a stopping time. Then the random variable

X

T

(!) = X

T (!)(!)

is FT

-measurable.


Martingales

We assume that {Ft

, t � 0} is a filtration.

DefinitionAn adapted process M = {M

t

, t � 0} is called a martingale with respect to Ft

if

(i) For all t � 0, E(|Mt

|) < 1.

(ii) For each s t , E(Mt

|Fs

) = M

s

.

Property (ii) can also be written as :

E(Mt

� M

s

|Fs

) = 0

M

t

is a supermartingale (or submartingale) if property (ii) is replaced byE(M

t

|Fs

) M

s

(or E(Mt

|Fs

) � M

s

).


Martingales

We assume that {Ft



t


if


|) < 1.


|Fs

) = M

s

.


E(Mt

� M

s

|Fs

) = 0

M

t


t

|Fs

) M

s

(or E(Mt

|Fs

) � M

s

).


Martingales

We assume that {Ft



t


if


|) < 1.


|Fs

) = M

s

.


E(Mt

� M

s

|Fs

) = 0

M

t


t

|Fs

) M

s

(or E(Mt

|Fs

) � M

s

).


Basic properties

1. For any integrable random variable X , {E(X |Ft

), t � 0} is a martingale.

2. If M

t

is a submartingale, then t ! E [Mt

] is nondecreasing.

3. If M

t

is a martingale and ' is a convex function such thatE(|'(M

t

)|) < 1 for all t � 0, then '(Mt

) is a submartingale.Proof : By Jensen’s inequality, if s t ,

E ('(Mt

)|Fs

) � ' (E(Mt

|Fs

)) = '(Ms

). ⇤

In particular, if M

t

is a martingale such that E(|Mt

|p) < 1 for all t � 0and for some p � 1, then |M

t

|p is a submartingale.


Examples :

Let B

t

be a Brownian motion Ft

the filtration generated by B

t

:

Ft

= �{B

s

, 0 s t}.

Then, the processes

M

(1)t

= B

t

M

(2)t

= B

2t

� t

M

(3)t

= exp(aB

t

� a

2t

2)

where a 2 R, are martingales.


1. B

t

is a martingale because

E(Bt

� B

s

|Fs

) = E(Bt

� B

s

) = 0.

2. For B

2t

� t , we can write, using the properties of the conditionalexpectation, for s < t

E(B2t

|Fs

) = E((Bt

� B

s

+ B

s

)2 |Fs

)

= E((Bt

� B

s

)2 |Fs

) + 2E((Bt

� B

s

)B

s

|Fs

)

+E(B2s

|Fs

)

= E (Bt

� B

s

)2 + 2B

s

E((Bt

� B

s

) |Fs

) + B

2s

= t � s + B

2s

.

3. Finally, for exp(aB

t

� a

2t

2 ) we have

E(eaB

t

� a

2t

2 |Fs

) = e

aB

s

E(ea(Bt

�B

s

)� a

2t

2 |Fs

)

= e

aB

s

E(ea(Bt

�B

s

)� a

2t

2 )

= e

aB

s

e

a

2(t�s)2 � a

2t

2 = e

aB

s

� a

2s

2 .


1. B

t


E(Bt

� B

s

|Fs

) = E(Bt

� B

s

) = 0.

2. For B

2t


E(B2t

|Fs

) = E((Bt

� B

s

+ B

s

)2 |Fs

)

= E((Bt

� B

s

)2 |Fs

) + 2E((Bt

� B

s

)B

s

|Fs

)

+E(B2s

|Fs

)

= E (Bt

� B

s

)2 + 2B

s

E((Bt

� B

s

) |Fs

) + B

2s

= t � s + B

2s

.


t

� a

2t

2 ) we have

E(eaB

t

� a

2t

2 |Fs

) = e

aB

s

E(ea(Bt

�B

s

)� a

2t

2 |Fs

)

= e

aB

s

E(ea(Bt

�B

s

)� a

2t

2 )

= e

aB

s

e

a

2(t�s)2 � a

2t

2 = e

aB

s

� a

2s

2 .


1. B

t


E(Bt

� B

s

|Fs

) = E(Bt

� B

s

) = 0.

2. For B

2t


E(B2t

|Fs

) = E((Bt

� B

s

+ B

s

)2 |Fs

)

= E((Bt

� B

s

)2 |Fs

) + 2E((Bt

� B

s

)B

s

|Fs

)

+E(B2s

|Fs

)

= E (Bt

� B

s

)2 + 2B

s

E((Bt

� B

s

) |Fs

) + B

2s

= t � s + B

2s

.


t

� a

2t

2 ) we have

E(eaB

t

� a

2t

2 |Fs

) = e

aB

s

E(ea(Bt

�B

s

)� a

2t

2 |Fs

)

= e

aB

s

E(ea(Bt

�B

s

)� a

2t

2 )

= e

aB

s

e

a

2(t�s)2 � a

2t

2 = e

aB

s

� a

2s

2 .


Optional Stopping Theorem

Theorem (Optional Stopping Theorem)Suppose that M

t

is a continuous martingale and let S T K two

bounded stopping times. Then

E(MT

|FS

) = M

S

.

This theorem implies that E(MT

) = E(MS

).In the submartingale case we have E(M

T

|FS

) � M

S

.As a consequence, if T is a bounded stopping time,

M

t

(sub)martingale ) M

t^T

(sub)martingale


Proof :

We will show that E(MT

) = E(M0).

Assume first that T takes value in a finite set :

0 t1 · · · t

n

K .

Then, by the martingale property

E(MT

) =n

X

i=1

E(MT

1{T=t

i

}) =n

X

i=1

E(Mt

i

1{T=t

i

})

=n

X

i=1

E(Mt

n

1{T=t

i

}) = E(Mt

n

) = E(M0).

In the general case we approximate T by the following nonincreasingsequence of stopping times

⌧n

=2n

X

k=1

kK

2n

1{ (k�1)K2n

T< kK

2n

}.


By continuityM⌧

n

a.s.! M

T

.

To show that E(M0) = E(M⌧n

) ! E(MT

), it suffices to check that thesequence M⌧

n

is uniformly integrable. This follows from :

E(|M⌧n

|1{|M⌧n

|�A}) =2n

X

k=1

E(|MkK

2n

|1{|MkK

2n

|�A,⌧n

= kK

2n

})

2n

X

k=1

E(|MK

|1{|MkK

2n

|�A,⌧n

= kK

2n

})

= E(|MK

|1{|M⌧n

|�A})

E(|MK

|1{sup0sK

|Ms

|�A}),

which converges to zero as A " 1, uniformly in n. ⇤


Doob’s maximal inequalities

TheoremLet {M

t

, t 2 [0,T ]} be a continuous martingale such that E(|MT

|p) < 1 for

some p � 1. Then, for all � > 0 we have

P

sup0tT

|Mt

| > �

!

1�p

E(|MT

|p). (1)

If p > 1, then

E

sup0tT

|Mt

|p!

✓

p

p � 1

◆

p

E(|MT

|p). (2)


Proof of (1) :

Set⌧ = inf{s � 0 : |M

s

| � �} ^ T .

Because ⌧ is a bounded stopping time and |Mt

|p is a submartingale,

E(|M⌧ |p) E(|MT

|p).

From the definition of ⌧ ,

|M⌧ |p � 1{sup0tT

|Mt

|��}�p + 1{sup0tT

|Mt

|<�}|MT

|p.

Therefore,

P

sup0tT

|Mt

| > �

!

1�p

E(|M⌧ |p) 1�p

E(|MT

|p).


Application to Brownian hitting times

Let B

t

be a Brownian motion. Fix a 2 R and consider the hitting time

⌧a

= inf{t � 0 : B

t

= a}

PropositionIf a < 0 < b, then

P(⌧a

< ⌧b

) =b

b � a

.

Proof : By the optional stopping theorem

E(Bt^⌧

a

) = E(B0) = 0.

Letting t ! 1 and using the dominated convergence theorem, it follows that

0 = aP(⌧a

< ⌧b

) + b(1 � P(⌧a

< ⌧b

)). ⇤


PropositionLet T = inf{t � 0 : B

t

/2 (a, b)}, where a < 0 < b. Then

E(T ) = �ab.

Proof : Using that B

2t

� t is a martingale, we get

E(B2T^t

) = E(T ^ t).

Therefore,E(T ) = lim

t!1E(B2

T^t

) = E(B2T

) = �ab. ⇤


PropositionFix a > 0. The hitting time

⌧a

= inf{t � 0 : B

t

= a},

satisfies

E [exp (�↵⌧a

)] = e

�p

2↵a. ↵ > 0 (3)


Proof :

For any � > 0, the process M

t

= e

�B

t

��2t

2 is a martingale such that

E(Mt

) = E(M0) = 1.

By the optional stopping theorem we obtain, for all N � 1.

E(M⌧a

^N

) = 1.

Notice that M⌧a

^N

= exp⇣

�B⌧a

^N

� �2(⌧a

^N)2

⌘

e

a�. So, by thedominated convergence theorem we obtain

E (M⌧a

) = 1,

that is,

E

✓

exp✓

��2⌧a

2

◆◆

= e

��a.

With the change of variables �2

2 = ↵, we get

E ( exp (�↵⌧a

)) = e

�p

2↵a. ⇤ (4)


The expectation of ⌧a

can be obtained by computing the derivative of (4)with respect to the variable ↵ :

E ( ⌧a

exp (�↵⌧a

)) =ae

�p

2↵a

p2↵

,

and letting ↵ # 0 we obtain E(⌧a

) = +1.

We can compute the density function of ⌧a

:

f⌧a

(s) =ap2⇡

s

� 32e

�a

2/2s, s � 0.


Strong Markov property

TheoremLet B be a Brownian motion and let T be a finite stopping time with respect to

the filtration FB

t

generated by B. Then the process

{B

T+t

� B

T

, t � 0}

is a Brownian motion independent of B

T

.

As a consequence, for any f 2 C

b

(R) and any finite stopping time T forthe filtration FB

t

, we have

E [f (BT+t

)|FB

T

] = (Pt

f )(BT

),

where P

t

is the semigroup associated with the Brownian motion B.


Proof :

Consider the process B

t

= B

T+t

� B

T

and suppose first that T isbounded. Let � 2 R and 0 s t . Applying the optional stoppingtheorem to the martingale

exp✓

i�B

t

+�2

t

2

◆

,

yieldsE

h

e

i�B

T+t

+�22 (T+t)|F

T+s

i

= e

i�B

T+s

+�22 (T+s).

Therefore,E

h

e

i�(BT+t

�B

T+s

)|FT+s

i

= e

��22 (t�s).

This implies that the increments of B are independent, stationary andnormally distributed.

If T is not bounded, then we can consider the stopping time T ^ N andlet N ! 1.


Reflection principle

TheoremLet M

t

= sup0st

B

s

. Then

P(Mt

� a) = 2P(Bt

> a) = 21p2⇡t

Z 1

a

e

� x

22

dx .


Proof :

We have

P(Bt

� a) = P(Bt

� a,Mt

� a) = P(Bt

� a|Mt

� a)P(Mt

� a)

= P(Bt

� a|⌧a

t)P(Mt

� a).

We know that {B⌧a

+s

� a, s � 0} is a Brownian motion independent ofF⌧

a

. Therefore,

P(Bt

� a|⌧a

t) = E [P(B⌧a

+(t�⌧a

) � a � 0|F⌧a

)|⌧a

t ] =12.


Brownian filtration

DefineFB

t

= � {B

s

, 0 s t} .

Denote by N the family of sets inF of probability zero (null sets).

PropositionThe filtration

Ft

= �n

FB

t

,No

.

is right-continuous. Therefore, it satisfies conditions (i) and (ii).


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Lecture 1: Brownian motion, martingales and Markov processessong/Gene Golub Summer School...Lecture 1: Brownian motion, martingales and Markov processes David Nualart Department of