stochastic differential equation

stochastic differential equation

Consider the

ordinary differential equation

, for
example, the population growth model

d

⁢

X

⁢

(

t

)

d

⁢

t

=

a

⁢

(

t

)

⁢

X

⁢

(

t

)

,

X

⁢

(

0

)

=

X

0

,

where

a

⁢

(

t

)

is the relative rate of growth at time

t

, and

X

⁢

(

t

)

is the solution-trajectory
of the system.

But we may want to take into account, in our model,
the randomness or the uncertainty
of our knowledge of the data.
In this case we may introduce the data

a

⁢

(

t

)

as:

a

⁢

(

t

)

=

r

⁢

(

t

)

+

N

⁢

(

t

)

,

where

N

⁢

(

t

)

is a noise term, represented by a

random variable

with some postulated probability distribution.

In general,

stochastic differential equations

can be posed in the case that the

infinitesimal

increment

d

⁢

X

⁢

(

t

)

is a

Gaussian

random variable. (Other types of random variables are
also possible, but require

extensions

of the basic theory.)
A

stochastic differential equation

(SDE) is an equation
of the form:

d

⁢

X

⁢

(

t

;

ω

)

=

μ

⁢

(

t

;

ω

)

⁢

d

⁢

t

+

σ

⁢

(

t

;

ω

)

⁢

d

⁢

W

⁢

(

t

;

ω

)

where

ω

lives in some

probability space

, and

W

⁢

(

t

)

is a

Wiener process

on that probability space.
The real-valued functions

μ

and

σ

are to

satisfy

certain measurability requirements, and are usually assumed to be known, with the process

X

⁢

(

t

)

being sought.

The

argument

ω

is usually suppressed in the notation:

d

⁢

X

⁢

(

t

)

=

μ

⁢

(

t

)

⁢

d

⁢

t

+

σ

⁢

(

t

)

⁢

d

⁢

W

⁢

(

t

)

,

(1)

with the understanding that

X

⁢

(

t

)

,

W

⁢

(

t

)

,

μ

⁢

(

t

)

and

σ

⁢

(

t

)

denote
random variables for each time

t

.

The

interpretation

of
the stochastic differential equation (

1

) is that
a process

X

⁢

(

t

)

satisfies it if and only if it satisfies
this

relation

amongst integrals:

X

⁢

(

t

1

)

-

X

⁢

(

t

0

)

=

∫

t

0

t

1

μ

⁢

(

t

)

⁢

𝑑

t

+

∫

t

0

t

1

σ

⁢

(

t

)

⁢

𝑑

W

⁢

(

t

)

(2)

for all times

t

0

and

t

1

.
The last integral is an Itô integral.

In many cases, the coefficients

μ

and

σ

depend on

X

⁢

(

t

)

itself:

d

⁢

X

⁢

(

t

)

=

μ

⁢

(

t

,

X

⁢

(

t

)

)

⁢

d

⁢

t

+

σ

⁢

(

t

,

X

⁢

(

t

)

)

⁢

d

⁢

W

⁢

(

t

)

.

In this case, equation (

2

) does not give
an explicit solution for the stochastic differential equation.
Nevertheless, there are

theorems

analogous to those
of ordinary differential equations,
that guarantee

existence

of solutions given certain
bounds on the growth of the coefficients

μ

⁢

(

t

,

x

)

and

σ

⁢

(

t

,

x

)

.

In simpler cases, stochastic differential equations that
involve unknowns on the right-hand side may still be solved
explicitly using changes of variables (often called Itô’s

formula

in this

context

).
For example,

X

⁢

(

t

)

=

X

0

⁢

e

-

κ

⁢

t

+

θ

⁢

(

1

-

e

-

κ

⁢

t

)

+

σ

⁢

∫

0

t

e

-

κ

⁢

(

t

-

s

)

⁢

𝑑

W

⁢

(

s

)

(for any

initial condition

X

0

) provides a solution to:

d

⁢

X

⁢

(

t

)

=

κ

⁢

(

θ

-

X

⁢

(

t

)

)

⁢

d

⁢

t

+

σ

⁢

d

⁢

W

⁢

(

t

)

.

References

1

Bernt Øksendal.

,
An

Introduction

with Applications.

5th ed. Springer 1998.

2

Lawrence Evans.

.

Department of Mathematics,
U.C. Berkeley.

Title

stochastic differential equation

Canonical name

StochasticDifferentialEquation

Date of creation

2013-03-22 16:10:07

Last modified on

2013-03-22 16:10:07

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

13

Author

stevecheng (10074)

Entry type

Definition

Classification

msc 60H10

Classification

msc 34-00

Synonym

SDE

Related topic

ItoIntegral

Related topic

WienerProcess

Related topic

BrownianMotion

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