The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 179 Wednesday, February 4, 2004 3:00 PM

Concepts of Probability 179

When it is necessary to emphasize the dependence of the random

variable from both time t and the element ω, a stochastic process is

explicitly written as a function of two variables: X = X(t,ω). Given ω,

the function X = Xt(ω) is a function of time that is referred to as the

path of the stochastic process.

The variable X might be a single random variable or a multidimen-

sional random vector. A stochastic process is therefore a function X =

X(t,ω) from the product space [0,T] × Ω into the n-dimensional real space

Rn. Because to each ω corresponds a time path of the process—in general

formed by a set of functions X = Xt(ω)—it is possible to identify the space

Ω with a subset of the real functions defined over an interval [0,T].

Let’s now discuss how to represent a stochastic process X = X(t,ω)

and the conditions of identity of two stochastic processes. As a stochas-

tic process is a function of two variables, we can define equality as

pointwise identity for each couple (t,ω). However, as processes are

defined over probability spaces, pointwise identity is seldom used. It is

more fruitful to define equality modulo sets of measure zero or equality

with respect to probability distributions. In general, two random vari-

ables X,Y will be considered equal if the equality X(ω) = Y(ω) holds for

every ω with the exception of a set of probability zero. In this case, it is

said that the equality holds almost everywhere (denoted a.e.).

A rather general (but not complete) representation is given by the

finite dimensional probability distributions. Given any set of indices

t 1 ,...,tm, consider the distributions

μt H [(

1 , ..., t

() = PXt

1

, ..., Xt ) ∈ H, H ∈ Bn ]

m m

These probability measures are, for any choice of the ti, the finite-

dimensional joint probabilities of the process. They determine many,

but not all, properties of a stochastic process. For example, the finite

dimensional distributions of a Brownian motion do not determine

whether or not the process paths are continuous.

In general, the various concepts of equality between stochastic pro-

cesses can be described as follows:

■ Property 1. Two stochastic processes are weakly equivalent if they have

the same finite-dimensional distributions. This is the weakest form of

equality.

■ Property 2. The process X = X(t,ω) is said to be equivalent or to be a

modification of the process Y = Y(t,ω) if, for all t,