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