The Mathematics of Financial Modelingand Investment Management

10-StochDiffEq Page 279 Wednesday, February 4, 2004 12:51 PM

Stochastic Differential Equations 279

The simplest example of a linear stochastic equation is the equation

of an arithmetic Brownian motion with drift, written as follows:

dXt = μdt + σdBt , 0 ≤t < ∞

X 0 = ξ, μ, σconstants

In linear equations of this type, the stochastic part enters only in an

additive way through the terms σij(t)dBt. The functions σ(t) are some-

times called the instantaneous variances and covariances of the process.

In the example of the arithmetic Brownian motion, μis called the drift

of the process and σthe volatility of the process.

It is intuitive that the solution of this equation is given by the solu-

tion of the associated deterministic equation, that is, the ordinary differ-

ential equation obtained by removing the stochastic part, plus the

cumulated random disturbances. Let’s first consider the associated

deterministic differential equation

dx

-------= At()xa t+ (), 0 ≤t < ∞

dt

where x(t) is a d-dimensional vector with initial conditions x(0) = ξ.

It can be demonstrated that this equation has an absolutely continu-

ous solution in the domain 0 ≤t < ∞. To find its solution, let’s first con-

sider the matrix differential equation

dΦ

--------= At()Φ, 0 ≤t < ∞

dt

This matrix differential equation has an absolutely continuous solution

in the domain 0 ≤t < ∞. The matrix Φ(t) that solves this equation is

called the fundamental solution of the equation. It can be demonstrated

that Φ(t) is a nonsingular matrix for each t. Lastly, it can be demon-

strated that the solution of the equation:

dx

-------= At()xa t+ (), 0 ≤t < ∞

dt

with initial condition x(0) = ξ, can be written in terms of the fundamen-

tal solution as follows: