The Mathematics of Financial Modelingand Investment Management

10-StochDiffEq Page 280 Wednesday, February 4, 2004 12:51 PM

280 The Mathematics of Financial Modeling and Investment Management

t

xt()= Φ()t x () 0 + ∫Φ–^1 ()sas()sd , 0 ≤t < ∞

0

Let’s now go back to the stochastic equation

dXt = [At()Xt + at()]dt + σ()tdBt , 0 ≤t < ∞

X 0 = ξ

Using Itô’s formula, it can be demonstrated that the above linear sto-

chastic equation admits the following unique solution:

t t

Xt()= Φ()ξ t + ∫ Φ–^1 ()sas()sd + ∫Φ–^1 ()σs ()sdBs , 0 ≤t < ∞

0 0

This effectively demonstrates that the solution of the linear stochastic

equation is the solution of the associated deterministic equation plus the

cumulated stochastic term

t

∫Φ s ()dB

• 1
  ()σs s
  0

To illustrate this, below we now specialize the above solutions in the

case of arithmetic Brownian motion, Ornstein-Uhlenbeck processes, and

geometric Brownian motion.

The Arithmetic Brownian Motion

The arithmetic Brownian motion in one dimension is defined by the fol-

lowing equation:

dXt = μdt + σdBt

In this case, A(t) = 0, a(t) = μ, σ(t) = σand the solution becomes

X = μt + σB

The Ornstein-Uhlenbeck Process

The Ornstein-Uhlenbeck process in one dimension is a mean-reverting

process defined by the following equation: