What Is a First-Passage Process?

This book is concerned with the first-passage properties of random walks and diffusion, and the basic consequences of first-passage phenomena. Our starting point is the first-passage probability; this is the probability that a diffusing particle or a random-walk first reaches a specified site (or set of sites) at a specified time. The importance of first-passage phenomena stems from its fundamental role in stochastic processes that are triggered by a first passage event. Typical examples include fluorescence quenching, in which light emission by a fluorescent molecule stops when it reacts with a quencher; integrate-and-fire neurons, in which a neuron fires only when a fluctuating voltage level first reaches a specified level; and the execution of buy/sell orders when a stock price first reaches a threshold. Further illustrations are provided throughout this book.

A Simple Illustration

To appreciate the essential features of first-passage phenomena, we begin with a simple example. Suppose that you are a nervous investor who buys stock in a company at a price of $100. Suppose also that this price fluctuates daily in a random multiplicative fashion. That is, at the end of each day the stock price changes by a multiplicative factor f < 1 or by f−1 compared with the previous day's price, with each possibility occurring with probability 1/2 (Fig. 1.1). The multiplicative change ensures that the price remains positive. To be concrete, let's take f = 90% and suppose that there is a loss on the first day so that the stock price drops to $90.