Stochastic Processes

Stochastic processes provide a language for systems that evolve under uncertainty. Probability sits at the meeting point of several parts of mathematics, from analysis and algebra through combinatorics, statistics, and mathematical modeling. A deep study of stochastic processes eventually requires measure theory, functional analysis, and more advanced probability than we develop in this undergraduate-level text. The aim here is to be self-contained, to develop essential ideas carefully, and to show how mathematics grows from questions arising in applications.

The text grew from lectures and courses given at Western Kentucky University, Cal Poly Pomona, The Australian National University, and the Claremont Colleges. The book assumes a first calculus-based course in probability and some comfort with elementary matrix calculations, but it does not assume that the reader already knows the language of filtrations, stopping times, generators, or stationary processes. Those ideas are introduced when they are needed, with reminders and supporting material in the appendices. The result can be used for a one-semester course, a reading course, or independent study. The exercises begin with computation and move toward proof, interpretation, and model construction; the appendix provides an answer, a hint, or a solution for every exercise.

The chapters are arranged in a principal sequence, but stochastic-process theory is not a single chain of prerequisites. Chapter 1 supplies the common probability language. Finite chains and first-step methods then lead to random walks and branching processes, while event-time models lead to continuous-time Markov chains and population processes. Reliability, risk, and epidemic models draw on several of these strands at once. The second-order and signal chapters take a different analytical route, beginning with mean-square geometry rather than a state transition rule.