Fundamentals of Actuarial Mathematics, 3rd Edition

18 An introduction to stochastic processes

18.1 Introduction

The purpose of this chapter is to provide background for the remaining chapters in the book. We make much use of the concept of conditioning, so the reader may wish to review Sections A.2 and A.8 of Appendix A.

A stochastic process is the tool used to model a quantity varying randomly in time. The following are the essential ingredients. We have an index set T, which gives the points of time that we are interested in. Normally, T will either be the nonnegative integers, 0, 1, 2, … (discrete time) or the whole nonnegative line [0, ∞) (continuous time). In both cases, we will sometimes have a maximum time horizon that we are interested in, in which case, ∞ will be replaced by a finite N). We also need a sample space with a probability measure P, and for each t in T, a random variable X_t defined on this space. The random variable X_t gives the value at time t, of the random quantity that we are trying to model. A stochastic process can then be defined formally as a collection of random variables X_t defined for each t in a set T.

We will illustrate briefly by considering the price of a certain stock. Anyone who has been involved with the stock market can attest that this is indeed a quantity that varies in time, and is subject to all kinds of random influences. Let time be discrete and refer to days. Suppose the stock is selling for 100 per share now, and we know that each day it will either increase by 20% or decrease ...