Essentials of Stochastic Processes

This book is for a first course on stochastic processes to be taken by undergraduates or master's students, who have had a course in probability theory, but who have not had a course in measure theory. It covers Markov chains in discrete and continuous time, Poisson processes, renewal theory, and Brownian motion and martingales. The last two topics are important for the brief treatment of option pricing. The book presents only the essentials of the subject, the parts of the theory most important for applications. To allow readers to choose their own level of detail, many of the proofs begin with a nonrigorous answer to the question "Why is this true ?" followed by a proof that fills in the missing details. Probability theory was developed to solve problems, so most of the text is devoted to analyzing examples. There are more than 325 carefully chosen problems to deepen the reader's understanding.