BROWNIAN MOTIONA tutorial

Krzysztof BurdzyUniversity of Washington

A paradox

|)(|sup,]1,0[:]1,0[

tfRft

(*)))((2

1exp)(

)10,)()((1

0

2

dttfc

ttfBtfP t

(*) is maximized by f(t) = 0, t>0

Microsoft stock- the last 5 years

The most likely (?!?) shape of a Brownian path:

Definition of Brownian motion

Brownian motion is the unique process with the following properties:(i) No memory(ii) Invariance(iii) Continuity (iv) tBVarBEB tt )(,0)(,00

Memoryless process

0t2t1t 3t

,,,231201 tttttt BBBBBB

are independent

Invariance

The distribution ofdepends only on t.

sst BB

Path regularity

(i) is continuous a.s.

(ii) is nowhere differentiable a.s.tBt

tBt

Why Brownian motion?Brownian motion belongs to several families

of well understood stochastic processes:

(i) Markov processes

(ii) Martingales

(iii) Gaussian processes

(iv) Levy processes

Markov processes

}0,|,{}|,{ suBstBBstB utst LL

The theory of Markov processes uses tools from several branches of analysis:(i) Functional analysis (transition semigroups)(ii) Potential theory (harmonic, Green functions)(iii) Spectral theory (eigenfunction expansion)(iv) PDE’s (heat equation)

Martingales are the only family of processesfor which the theory of stochastic integrals is fully developed, successful and satisfactory.

Martingales

sst BBBEts )|(

t

ss dBX0

Gaussian processes

nttt BBB ,,,21

is multidimensionalnormal (Gaussian)

(i) Excellent bounds for tails(ii) Second moment calculations(iii) Extensions to unordered parameter(s)

The Ito formula

nt

knknknk

n

t

ss BBXdBX0

//)1(/

0

)(lim

t t

ssst dsBfdBBfBfBf0 0

0 )(2

1)()()(

Random walk

Independent steps, P(up)=P(down)

0,0,/ tBtWa ta

at

(in distribution)

tW

t

Scaling

Central Limit Theorem (CLT), parabolic PDE’s

}10,{}10,{ / tBatB at

D

t

The effect is the same as replacing with

Multiply the probability of each Brownian path by

Cameron-Martin-Girsanov formula

}10,{ tBt

1

0

1

0

2))((2

1)(exp dssfdBsf s

}10,{ tBt }10),({ ttfBt

Invariance (2)

}10,{}10,{ 11 tBBtB t

D

t

Time reversal

0 1

Brownian motion and the heat equation),( txu – temperature at location x at time t

),(2

1),( txutxu

t x

Heat equation:

dxxudx )0,()(

)(),( dyBPdytyu t Forward representation

)0,(),( yBEutyu t Backward representation(Feynman-Kac formula)

0 t

y

Multidimensional Brownian motion

,,, 321ttt BBB - independent 1-dimensional

Brownian motions

),,,( 21 dttt BBB - d-dimensional Brownian

motion

Feynman-Kac formula (2)

)(xf

0

)(exp)()( dsBVBfExu sx

B x

0)()()(2

1 xuxVxu

Invariance (3)

The d-dimensional Brownian motion is invariantunder isometries of the d-dimensional space.It also inherits invariance properties of the1-dimensional Brownian motion.

)2/)(exp(2

1

)2/exp(2

1)2/exp(

2

1

22

21

22

21

xx

xx

Conformal invariance

f

}0),()({ 0 tBfBf t

analytictB )( tBf

has the same distribution as

t

stc dsBftctB0

2)( |)(|)(},0,{

If then

The Ito formula Disappearing terms (1)

t t

ssst dsBfdBBfBfBf0 0

0 )(2

1)()()(

t

sst dBBfBfBf0

0 )()()(

0f

Brownian martingales

Theorem (Martingale representation theorem).{Brownian martingales} = {stochastic integrals}

},{,)|(

0

tsBFMMMME

dBXM

sBttsst

t

sst

The Ito formula Disappearing terms (2)

t

s

t

s

t

sst

dsBsfx

dsBsfs

dBBsfx

BtfBtf

02

2

0

0

0

),(2

1),(

),(),(),(

t

s

t

s

t

dsBsfx

EdsBsfs

E

BtEfBtEf

02

2

0

0

),(2

1),(

),(),(

Mild modifications of BM

Mild = the new process corresponds to the Laplacian(i) Killing – Dirichlet problem(ii) Reflection – Neumann problem(iii) Absorption – Robin problem

Related models – diffusions

dtXdBXdX tttt )()(

(i) Markov property – yes(ii) Martingale – only if(iii) Gaussian – no, but Gaussian tails

0

Related models – stable processes

2/1)(dtdB /1)(dtdX

Brownian motion –Stable processes –

(i) Markov property – yes(ii) Martingale – yes and no(iii) Gaussian – no

Price to pay: jumps, heavy tails, 20 220

Related models – fractional BM/1)(dtdX

(i) Markov property – no(ii) Martingale – no(iii) Gaussian – yes (iv) Continuous

1 21

Related models – super BM

Super Brownian motion is related to2uu

and to a stochastic PDE.

Related models – SLE

Schramm-Loewner Evolution is a model for non-crossing conformally invariant2-dimensional paths.

(i) is continuous a.s.

(ii) is nowhere differentiable a.s.

(iii) is Holder

(iv) Local Law if Iterated Logarithm

Path properties

tBt

tBt

tBt )2/1(

1|log|log2

suplim0

tt

Btt

Exceptional points

1|log|log2

suplim0

tt

Btt

1|)log(|log)(2

suplim

stst

BB st

st

For any fixed s>0, a.s.,

),0()(2

suplim

st

BB st

st

There exist s>0, a.s., such that

Cut points

For any fixed t>0, a.s., the 2-dimensionalBrownian path contains a closed looparound in every interval tB ),( ttAlmost surely, there exist such that

)1,0(t ])1,(()),0([ tBtB

Intersection properties

1))((.,.)4(

2))((.,.

2))((.,.)3(

))((.,.

..)2(

.,.)1(

14

13

13

12

xBCardRxsad

xBCardRxsa

xBCardRxsad

xBCardRxsa

BBtssatd

BBtstsad

ts

ts

Intersections of random sets

BA

dBA

)dim()dim(

The dimension of Brownian trace is 2in every dimension.

Invariance principle

(i) Random walk converges to Brownianmotion (Donsker (1951))(ii) Reflected random walk convergesto reflected Brownian motion(Stroock and Varadhan (1971) - domains,B and Chen (2007) – uniform domains, not alldomains)(iii) Self-avoiding random walk in 2 dimensionsconverges to SLE (200?) (open problem)

2C

Local time

sts

t

Bt

BM

dsLs

sup

2

1lim }{

0

t

0

1

Local time (2)

}10,|{|}10,{

}10,{}10,{

tBtBM

tMtL

t

D

tt

t

D

t

Local time (3)

}{inf0

tLss

t

Inverse local time is a stable processwith index ½.

References

R. Bass Probabilistic Techniques in Analysis, Springer, 1995

F. Knight Essentials of Brownian Motion and Diffusion, AMS, 1981

I. Karatzas and S. Shreve Brownian Motion and Stochastic Calculus, Springer, 1988