The Mathematics of Financial Modelingand Investment Management

10-StochDiffEq Page 281 Wednesday, February 4, 2004 12:51 PM

Stochastic Differential Equations 281

dXt = –αXtdt + σdBt

It is a mean-reverting process because the drift is pulled back to zero by

a term proportional to the process itself. In this case, A(t) = –α, a(t) = 0 ,

σ(t) = σ and the solution becomes

∫

t

X – α (ts– )

t = X 0 + e

• αt + σ e dB
  s
  0

The Geometric Brownian Motion

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

lowing equation:

dX = μXdt + σXdB

This equation can be easily reduced to the previous linear case by the

transformation:

Y = log X

Let’s apply Itô’s formula

∂g ∂g 1 ∂^2 g ∂g

dYt = ------+ ------a + ------------b^2 



dt + ------bdBt

∂t ∂x^2 ∂x^2  ∂x

where

∂g 1 ∂^2 g 1

gt x( , ) = logx, -----

∂g

• = 0 , ------= ---, --------- –= ------
  ∂t ∂t x^2
  ∂x
  2
  x

We can then verify that the logarithm of the geometric Brownian motion

becomes an arithmetic Brownian motion with drift

1

μ′ = μ – ---σ^2

2

The geometric Brownian motion evolves as a lognormal process: