Elementary Applications of Probability Theory

For the second edition, two additional chapters, Chapters 11 and 12, have been written. The added material should make the book suitable for two consecutive courses in elementary and intermediate applications of probability. The new material consists of an introduction to stochastic differential equations. It is hoped that this will be useful for applied mathematical modelling of the behaviour of many naturally occurring randomly fluctuating quantities. An attempt has been made to explain the material with a certain amount of rigour, but hopefully without so much detail that a practical understanding is impaired. The stochastic differential equations in this book are first order equations with an additional noise term. This added term usually contains a Gaussian 'white noise' so that the resulting solution is called a diffusion process. Chapter 11 starts with a brief reminder of the nature of ordinary determinis­ tic differential equations, followed by an explanation of the essential differences between deterministic and stochastic equations. These have been illustrated with data in neurophysiology and economics. There follows a thorough discussion of the properties of the standard Wiener process which forms a cornerstone of the theory, and a section on white noise which is a useful concept, especially for modelling. The simplest stochastic differential equations, being those of the Wiener process with drift, are then introduced.