Review of Stochastic Processes and Models by David Stirzaker
Stochastic Processes and Models is a textbook by David Stirzaker, a professor of mathematics at Oxford University. The book provides a concise and lucid introduction to simple stochastic processes and models, such as random walks, renewals, Markov chains, martingales, Brownian motion, and diffusion processes. It also covers some applications of stochastic calculus, such as option pricing and stochastic differential equations.
The book is intended for an undergraduate second course in probability for students of statistics, mathematics, finance and operational research. It contains numerous exercises, problems and solutions to help the reader test their understanding and practice their skills. The book assumes some familiarity with basic probability theory and calculus, but does not require any prior knowledge of stochastic processes or models.
The book is divided into nine chapters. The first chapter introduces some basic concepts and notation of stochastic processes, such as sample paths, filtrations, stopping times, and conditional expectations. The second chapter discusses random walks and their properties, such as recurrence, transience, hitting times, and limit theorems. The third chapter studies renewal processes and their applications, such as Poisson processes, queues, reliability theory, and branching processes. The fourth chapter introduces Markov chains and their characteristics, such as transition probabilities, classification of states, stationary distributions, ergodicity, and convergence theorems. The fifth chapter extends the theory of Markov chains to continuous time and state space, such as birth-death processes, Markov jump processes, and Kolmogorov equations. The sixth chapter explores martingales and their properties, such as optional stopping theorem, Doob's inequalities, convergence theorems, and martingale representation theorem. The seventh chapter presents the Wiener process as a model for Brownian motion and its properties, such as continuity, Gaussian increments, scaling property, reflection principle, and quadratic variation. The eighth chapter develops diffusion processes as generalizations of the Wiener process and their properties, such as stochastic differential equations (SDEs), Ito's lemma, Feynman-Kac formula, Girsanov's theorem, and Ornstein-Uhlenbeck process. The ninth chapter gives a brief account of the stochastic integral and its applications to option pricing using the Black-Scholes formula.
The book is well-written and clear in its exposition. The author uses examples and diagrams to illustrate the concepts and results. The exercises are varied and challenging, ranging from simple calculations to proofs of important theorems. The solutions are detailed and helpful. The book covers most of the topics that are essential for a solid foundation in stochastic processes and models. However, some topics that are relevant for more advanced courses or research are omitted or treated briefly, such as measure-theoretic probability theory, stochastic control theory, stochastic filtering theory, Levy processes, jump-diffusion models etc.
Overall, Stochastic Processes and Models is a valuable resource for anyone who wants to learn about simple stochastic processes and models. It is suitable for undergraduate students who have some background in probability theory and calculus. It is also useful for graduate students or researchers who need a quick review or reference on the subject. aa16f39245