Introduction

A stochastic process is a sequence of random variables. It involves a random (stochastic) element and a time element. Examples:

• the unfunded liability of a pension fund at successive valuations
• rainfall since midnight
• the number of cars passing a traffic census
• a general insurance company's surplus on a particular class of business, evaluated continuously
• stock market index, observed day by day (or minute by minute)
• a football team's points score, match by match
• the size of the hole in the ozone layer

A probabilistic model for a real-life random processes must be

• simple enough that it can be analysed mathematically
• complex enough to mirror the behaviour of the process.

We shall look at a number of models, considering cases where the quantity observed is discrete or where it is continuous. In this course we restrict attention to the cases where the time element is discrete, i.e. only takes values 0, 1, 2, ¼. (The exception to this is that we shall briefly consider the Poisson process.) Cases where the time element is continuous will be covered in the companion course, Continuous Stochastic Modelling.

The tools we use are taken from probability theory, like the Central Limit Theorem, and from other branches of mathematics like matrix algebra or difference equations.

In the first section we shall deal with general ideas about modelling and about simulation. Both of these themes will run through this module and its successor, and both will be developed by means of practical applications in subsequent sections.

Neither modelling nor simulation is specifically concerned with probability. The first serious mathematical work in the course deals with the random walk, also known as the drunkard's walk, a discrete time process. This will be extended to a consideration of Markov chains in discrete time.

We consider a number of applications of these processes, including some examples where the probabilities of various outcomes are not constant, but are permitted to vary with time, such as applications where probabilities of significant events are affected by the age of the individual under consideration.

Attention is then transferred to continuous time, where we look at the Poisson process and some of its generalisations.

Finally we turn our attention to Time Series, an area of statistical theory which deals with the analysis of a sequence of dependent observations taken at equally-spaced times. Here the theoretical emphasis is on studying patterns of covariance, but there will be a substantial practical aspect involving MINITAB.