Transcript rstut 6958
BROWNIAN MOTION
A tutorial
Krzysztof Burdzy
University of Washington
A paradox
f : [0,1] R,
sup | f (t ) |
t[ 0,1]
P ( f (t ) Bt f (t ) , 0 t 1)
11
2
c( ) exp ( f (t )) dt
20
(*)
(*) is maximized by f(t) = 0, t>0
The most likely (?!?) shape of a
Brownian path:
Microsoft stock
- the last 5 years
Definition of Brownian motion
Brownian motion is the unique process
with the following properties:
(i) No memory
(ii) Invariance
(iii) Continuity
(iv) B0 0, E ( Bt ) 0, Var( Bt ) t
Memoryless process
t0
t1
t2
t3
Bt1 Bt0 , Bt2 Bt1 , Bt3 Bt2 ,
are independent
Invariance
The distribution of Bt s Bs
depends only on t.
Path regularity
(i) t Bt
(ii) t Bt
is continuous a.s.
is nowhere differentiable a.s.
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
L {Bt , t s | Bs } L {Bt , t s | Bu , 0 u s}
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
s t E ( Bt | Bs ) Bs
Martingales are the only family of processes
for which the theory of stochastic integrals is
fully developed, successful and satisfactory.
t
X
0
s
dBs
Gaussian processes
Bt1 , Bt2 , , Btn
is multidimensional
normal (Gaussian)
(i) Excellent bounds for tails
(ii) Second moment calculations
(iii) Extensions to unordered parameter(s)
The Ito formula
t
X
dB
lim
s
s
0
n
t
nt
X
k 0
k/n
( B( k 1) / n Bk / n )
t
1
f ( Bt ) f ( B0 ) f ( Bs )dBs f ( Bs )ds
20
0
Random walk
Wt
t
Independent steps, P(up)=P(down)
aW
t/a
,t 0
a
(in distribution)
Bt , t 0
Scaling
Central Limit Theorem (CLT),
parabolic PDE’s
D
{Bt , 0 t 1} { a Bt / a , 0 t 1}
Cameron-Martin-Girsanov formula
Multiply the probability of each Brownian path
{Bt , 0 t 1} by
1
1
1
2
exp f ( s )dBs ( f ( s )) ds
20
0
The effect is the same as replacing
{Bt , 0 t 1} with {Bt f (t ), 0 t 1}
Invariance (2)
Time reversal
D
{Bt , 0 t 1} {B1t B1 , 0 t 1}
0
1
Brownian motion and the heat equation
u ( x, t ) – temperature at location x at time t
Heat equation:
(dx) u ( x,0)dx
1
u ( x , t ) x u ( x, t )
t
2
Forward
representation
Backward representation
(Feynman-Kac formula)
u ( y, t )dy P ( Bt dy )
u( y, t ) Eu( Bt y,0)
y
0
t
Multidimensional Brownian motion
1
t
2
t
3
t
B , B , B , - independent 1-dimensional
1
t
2
t
Brownian motions
d
t
( B , B , , B )
- d-dimensional Brownian
motion
Feynman-Kac formula (2)
B
x
f (x )
1
u ( x) V ( x)u ( x) 0
2
x
u ( x) E f ( B ) exp V ( Bs )ds
0
Invariance (3)
The d-dimensional Brownian motion is invariant
under isometries of the d-dimensional space.
It also inherits invariance properties of the
1-dimensional Brownian motion.
1
1
2
2
exp( x1 / 2)
exp( x2 / 2)
2
2
1
2
2
exp( ( x1 x2 ) / 2)
2
Conformal invariance
Bt
f
f ( Bt )
analytic
{ f ( Bt ) f ( B0 ), t 0}
has the same distribution as
t
{Bc (t ) , t 0}, c(t ) | f ( Bs ) | ds
2
0
The Ito formula
Disappearing terms (1)
t
t
1
f ( Bt ) f ( B0 ) f ( Bs )dBs f ( Bs )ds
20
0
If
f 0
then
t
f ( Bt ) f ( B0 ) f ( Bs )dBs
0
Brownian martingales
Theorem (Martingale representation theorem).
{Brownian martingales} = {stochastic integrals}
t
M t X s dBs
0
E ( M t | M s ) M s , M t Ft {Bs , s t}
B
The Ito formula
Disappearing terms (2)
f (t , Bt ) f (t , B0 )
f ( s, Bs )dBs
x
0
t
1
f ( s, Bs )ds 2 f ( s, Bs )ds
s
2 0 x
0
t
t
2
Ef (t , Bt ) Ef (t , B0 )
1
2
E
f ( s, Bs )ds E 2 f ( s, Bs )ds
s
2 0 x
0
t
t
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
dX t ( X t )dBt ( X t )dt
(i) Markov property – yes
(ii) Martingale – only if 0
(iii) Gaussian – no, but Gaussian tails
Related models – stable processes
Brownian motion –
Stable processes –
dB (dt )
1/
dX (dt )
1/ 2
(i) Markov property – yes
(ii) Martingale – yes and no
(iii) Gaussian – no
Price to pay: jumps, heavy tails,
0 2
022
Related models – fractional BM
1/
dX (dt )
(i) Markov property – no
(ii) Martingale – no
(iii) Gaussian – yes
(iv) Continuous
1
1 2
Related models – super BM
Super Brownian motion is related to
u u
2
and to a stochastic PDE.
Related models – SLE
Schramm-Loewner Evolution is a model
for non-crossing conformally invariant
2-dimensional paths.
Path properties
(i) t Bt
(ii)
(iii)
(iv)
is continuous a.s.
t Bt is nowhere differentiable a.s.
t Bt is Holder (1 / 2 )
Local Law if Iterated Logarithm
Bt
lim sup
1
t 0
2t log | log t |
Exceptional points
Bt
lim sup
1
t 0
2t log | log t |
For any fixed s>0, a.s.,
Bt Bs
lim sup
1
t s
2(t s) log | log( t s) |
There exist s>0, a.s., such that
Bt Bs
lim sup
(0, )
t s
2(t s)
Cut points
For any fixed t>0, a.s., the 2-dimensional
Brownian path contains a closed loop
around Bt in every interval (t , t )
Almost surely, there exist t (0,1)
such that B([0, t )) B((t ,1])
Intersection properties
(d 1) a.s., t s t
Bs Bt
(d 2) t a.s. s t
Bs Bt
1
a.s., x R Card ( B ( x))
2
1
(d 3) a.s., x R Card ( B ( x)) 2
3
1
a.s., x R Card ( B ( x)) 2
3
1
(d 4) a.s., x R Card ( B ( x)) 1
4
Intersections of random sets
dim( A) dim( B) d
A B
The dimension of Brownian trace is 2
in every dimension.
Invariance principle
(i) Random walk converges to Brownian
motion (Donsker (1951))
(ii) Reflected random walk converges
to reflected Brownian motion
2
(Stroock and Varadhan (1971) - C domains,
B and Chen (2007) – uniform domains, not all
domains)
(iii) Self-avoiding random walk in 2 dimensions
converges to SLE (200?)
(open problem)
Local time
t
1
Lt lim
1
ds
{
B
}
s
0 2
0
M t sup Bs
s t
Local time (2)
D
{Lt ,0 t 1} {M t ,0 t 1}
D
{M t Bt ,0 t 1} {| B |t ,0 t 1}
Local time (3)
t inf {Ls t}
s 0
Inverse local time is a stable process
with 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