The Mathematics of Financial Modelingand Investment Management

10-StochDiffEq Page 276 Wednesday, February 4, 2004 12:51 PM

276 The Mathematics of Financial Modeling and Investment Management

Note that the solution of a differential equation is a stochastic pro-

cess. Initial conditions must therefore be specified as a random variable

and not as a single value as for ordinary differential equations. In other

words, there is an initial value for each state. It is possible to specify a sin-

gle initial value as the initial condition of a stochastic differential equa-

tion. In this case the initial condition is a random variable where the

probability mass is concentrated in a single point.

We omit the detailed proof of the theorem of uniqueness and exist-

ence. Uniqueness is proved using the Itô isometry and the Lipschitz con-

dition. One assumes that there are two different solutions and then

demonstrates that their difference must vanish. The proof of existence

of a solution is similar to the proof of existence of solutions in the

domain of ordinary equations. The solution is constructed inductively

by a recursive relationship of the type

t t

X(k +^1 )(tω , ) = φ[sXk(sω , )] sd +

∫ψ[sX

k(sω , )]dB

∫ , , s

0 0

It can be shown that this recursive relationship produces a sequence of

processes that converge to the unique solution.

GENERALIZATION TO SEVERAL DIMENSIONS

The concepts and formulas established so far for Itô (and Stratonovich)

integrals and processes can be extended in a straightforward but often cum-

bersome way to multiple variables. The first step is to define a d-dimen-

sional Brownian motion.

Given a probability space (Ωℑ,,P) equipped with a filtration {ℑt}, a

d-dimensional standard Brownian motion Bt(ω), is a stochastic process

with the following properties:

■ Bt(ω) is a d-dimensional process defined over the probability space

(Ωℑ,,P) that takes values in Rd.

■ Bt(ω) has continuous paths for 0 ≤t ≤∞.

■ B 0 (ω) = 0.

■ Bt(ω) is adapted to the filtration ℑt.

■ The increments Bt(ω) – Bs(ω) are independent of the σ-algebra ℑs and

have a normal distribution with mean zero and covariance matrix (t –

s)Id, where Id is the identity matrix.