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: