An Elementary Introduction to Mathematical Finance

Brownian Motion

A Brownian motion is a collection of random variables X(t), t ≥ 0 that satisfy certain properties that we will momentarily present. We imagine that we are observing some process as it evolves over time. The index parameter t represents time, and X(t) is interpreted as the state of the process at time t. Here is a formal definition.

Definition The collection of random variables X(t), t ≥ 0 is said to be a Brownian motion with drift parameter μ and variance parameter σ2 if the following hold:

1. (a) X(0) is a given constant.

2. (b) For all positive y and t, the random variable X(t + y) – X(y) is independent of the the process values up to time y and has a normal distribution with mean μt and variance tσ2.

Assumption (b) says that, for any history of the process up to the present time y, the change in the value of the process over the next t time units is a normal random with mean μt and variance tσ2. Because any future value X(t + y) is equal to the present value X(y) plus the change in value X(t + y) – X(y), the assumption implies that it is only the present value of the process, and not any past values, that determines probabilities about future values.

An important property of Brownian motion is that X(t) will, with probability 1, be a continuous function of t.