Ornstein-Uhlenbeck process

Ornstein-Uhlenbeck process

Definition

The

Ornstein-Uhlenbeck

process is a

stochastic process

that

satisfies

the following

stochastic differential equation

:

d

⁢

X

t

=

κ

⁢

(

θ

-

X

t

)

⁢

d

⁢

t

+

σ

⁢

d

⁢

W

t

,

(1)

where

W

t

is a standard

Brownian motion

on

t

∈

[

0

,

∞

)

.

The

constant

parameters are:

•

κ

>

0

is the rate of mean reversion;

•

θ

is the long-term mean of the process;

•

σ

>

0

is the volatility or average magnitude, per square-root time,
of the random fluctuations that are modelled as Brownian motions.

Mean-reverting property

If we ignore the random fluctuations in the process
due to

d

⁢

W

t

, then we see that

X

t

has an overall drift
towards a mean value

θ

.
The process

X

t

reverts to this mean exponentially, at rate

κ

,
with a magnitude in direct proportion to the distance
between the current value of

X

t

and

θ

.

This can be seen by looking at the

solution

to the

ordinary differential equation

d

⁢

x

t

=

κ

⁢

(

θ

-

x

)

⁢

d

⁢

t

which is

θ

-

x

t

θ

-

x

0

=

e

-

κ

⁢

(

t

-

t

0

)

,

or

⁢

x

t

=

θ

+

(

x

0

-

θ

)

⁢

e

-

κ

⁢

(

t

-

t

0

)

.

(2)

For this reason, the

Ornstein-Uhlenbeck process

is also called a

mean-reverting process

,
although the latter name applies to other types
of stochastic processes exhibiting the same property as well.

Solution

The solution to the stochastic differential equation (

1

)
defining the Ornstein-Uhlenbeck process is, for any

0

≤

s

≤

t

, is

X

t

=

θ

+

(

X

s

-

θ

)

⁢

e

-

κ

⁢

(

t

-

s

)

+

σ

⁢

∫

s

t

e

-

κ

⁢

(

t

-

u

)

⁢

𝑑

W

u

.

where the integral on the right is the Itô integral.

For any fixed

s

and

t

, the

random variable

X

t

,

conditional

upon

X

s

, is normally distributed with

mean

=

θ

+

(

X

s

-

θ

)

⁢

e

-

κ

⁢

(

t

-

s

)

,

variance

=

σ

2

2

⁢

κ

⁢

(

1

-

e

-

2

⁢

κ

⁢

(

t

-

s

)

)

.

Observe that the mean of

X

t

is exactly
the value derived heuristically
in the solution (

2

) of the ODE.

The Ornstein-Uhlenbeck process is a time-homogeneous Itô diffusion.

Applications

The Ornstein-Uhlenbeck process is widely used for modelling
biological processes such as neuronal response,
and in mathematical finance,
the modelling of the dynamics of interest rates
and volatilities of asset prices.

References

1

Martin Jacobsen. “Laplace and the Origin of the Ornstein-Uhlenbeck Process”.
Bernoulli, Vol. 2, No. 3. (Sept. 1996), pp. 271 – 286.

2

Bernt Øksendal.

Stochastic Differential Equations,
An

Introduction

with Applications

, 5th edition. Springer, 1998.

3

Steven E. Shreve.

Stochastic Calculus for Finance II:
Continuous-Time Models

. Springer, 2004.

4

Sebastian Jaimungal. Lecture notes for

Pricing Theory

.
University of Toronto.

5

Dmitri Rubisov. Lecture notes for

Risk Management

.
University of Toronto.

Title

Ornstein-Uhlenbeck process

Canonical name

OrnsteinUhlenbeckProcess

Date of creation

2013-03-22 17:19:26

Last modified on

2013-03-22 17:19:26

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

4

Author

stevecheng (10074)

Entry type

Definition

Classification

msc 60H10

Classification

msc 60-00

Synonym

Ornstein-Uhlenbeck equation

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