AMS 216 Stochastic Differential Equations
Lecture #18
Short time asymptotics
We consider the stochastic differential equation
dX ( t ) = b ( X ) dt + 2a ( X )dW ( t )
Note that for mathematical convenience (which we will see below), we write the coefficie
AMS 216 Stochastic Differential Equations
Lecture #6
Now let us look at the general case.
t
f ( s, W ( s ) dW ( s ) ( Stratonovich )
0
= lim
N
= lim
N
N 1
(
( ) + f ( s
(
( ) + 1 ( f ( s
2
1
f sj, W sj
j=0 2
N 1
j +1
f sj, W sj
j=0
( ) dW
, W s j +1
j +1
AMS 216 Stochastic Differential Equations
Lecture #3
White noise
A long story (continued)
Fourier transform:
y(
= exp ( i2
) = F y (t )
y (t ) = F
y(
1
t ) y ( t ) dt
t ) y(
= exp ( i2
)
(Forward transform)
(Backward transform)
)d
Note that we introduced
AMS 216 Stochastic Differential Equations
Lecture #7
Backward equation and forward equation
Consider the stochastic differential equation
dX = b ( X, t ) dt + a ( X, t ) dW
(Ito interpretation)
Consider the transition probability density
q ( x,t z, s ) =
AMS 216 Stochastic Differential Equations
Lecture #8
Summary:
Backward equation:
We view the transition probability density q ( x,t z, s ) as a function of (z, s).
The backward equation is
1
a ( z, s ) qzz
2
q ( x,t z,t ) = ( x z )
qs = b ( z, s ) qz
The
AMS 216 Stochastic Differential Equations
Lecture #5
Hints on homework problems:
Exercises P1 and P2:
A method for solving difference equation
u ( n ) = a1u ( n + 1) + a2u ( n 1)
Consider solution of the form u ( n ) = r n .
Substituting into the differen
AMS 216 Stochastic Differential Equations
Lecture #11
First, I want to go over a homework problem
Exercise P5:
In the discussion of Ornstein-Uhlenbeck process, we obtained that
E cfw_Y (t ) = e
t
Y (0)
for t > 0
Is this still valid for -t < 0?
E cfw_Y ( t
AMS 216 Stochastic Differential Equations
Lecture #10
Breaking of a molecular bond (continued)
The full governing equation (Newtons second law)
dX = Y dt
bY dt
V ( X ) dt +
Viscous
drag
Force from
potential
mdY =
2kBT b dW
Brownian
force
In the limit of a
AMS 216 Stochastic Differential Equations
Lecture #9
Exit time (escape problem):
Consider the stochastic differential equation
dX = b( X ) dt + a( X ) dW
We study the problem of exiting (escaping from) a prescribed region.
Specifically, we study the time
AMS 216 Stochastic Differential Equations
Lecture #4
Exercise P3:
Consider the problem of generating samples of W(t) subject to boundary conditions.
Suppose we know W(t) = a and W(t+h) = b.
Find the conditional distribution of W t +
h
.
2
That is, to find
AMS 216 Stochastic Differential Equations
Lecture #12
Feynman-Kac formula (continued)
Now we derive the partial differential equation for u(x, t, T).
exp
T
t
( X ( s ), s ) ds
t
( x, t ) dt + o ( dt )
= exp
= (1
( x, t ) dt ) exp
cfw_
u ( x,t,T ) = E exp
AMS 216 Stochastic Differential Equations
Lecture #13
Jump process
We consider the stochastic differential equation
dX = b ( X ) dt + a ( X ) dW
It is a continuous space continuous time Markov process.
A jump process is a discrete space continuous time Ma
AMS 216 Stochastic Differential Equations
Lecture #16
Einstein-Smoluchowski limit (continued)
After scaling, the starting equations in the Einstein-Smoluchowski limit are
1
dX = Ydt
dY =
1
2
(Y
f ( X ) dt +
(E01)
1
2D0 dW
The target equation of the conver
AMS 216 Stochastic Differential Equations
Lecture #15
Einstein-Smoluchowski limit (continued)
The stochastic motion of a small particle in water
dX = Y dt
mdY = bY dt + F ( X ) dt + q dW
Claim:
As a
0, the stochastic motion is governed by
dX =
F(X)
dt + 2
AMS 216 Stochastic Differential Equations
Lecture #14
Now let us see an example of a discrete process converging to a continuous process.
Generalized Ornstein-Uhlenbeck process
Consider the stochastic differential equation
dY =
Y dt + d
Y ( 0 ) = y0
where
AMS 216 Stochastic Differential Equations
Lecture #1
Stochastic differential equation:
dX ( t ) = b ( X ( t ) ,t ) dt + a ( X ( t ) ,t ) dW ( t )
Or in a more concise form
dX = b( X,t) dt + a( X,t ) dW
We need to introduce W(t).
The Wiener process (Browni
AMS 216 Stochastic Differential Equations
Lecture #20
Short time asymptotics (continued)
a ( x ) = a0 , b ( x ) = b0 + b1 ( x
Case #2:
x0 )
We have
x
R( x) =
1
1
dy =
(x
a0
a0
x0
( y) =
R
x(s ) = R
b( x)
a( x)
0
R( x)
1
s
= (x
s
b0 + b1 ( x
x0 )
x0 )
b0 +
AMS 216 Stochastic Differential Equations
Lecture #17
Stochastic differential equations with population dependent driving force
Stochastic motion of a small particle in water (Einstein-Smoluchowski limit)
dX =
1
V ( X ) dt + 2D dW
b
where b is the drag co
AMS 216 Stochastic Differential Equations
Lecture #19
Short time asymptotics (continued)
Consider the Fokker-Planck equation
1
(b ( x ) p ) + ( a ( x ) p )
pt =
x
xx
The leading term of the solution is
( x, s )
p ( x, s ) = exp
A0 ( x, s )
1
2
( x,s) and
AMS 216 Stochastic Differential Equations
Homework assignments
Exercise #1:
Consider a biased game
dX = mdt + dW
Let T(x) denote the average time to the end of game
cfw_
T ( x ) = E time until X ( t ) = C or X ( t ) = 0 X ( 0 ) = x
We have T (0) = 0 and
AMS 216 Stochastic Differential Equations
Lecture #2
Gamblers Ruin (continued)
Question #1: How long can you play?
Question #2: What is the chance that you break the bank?
Note that unlike in the case of deterministic equations, for stochastic differentia