This course is an introduction to stochastic models for students in the Masters Program in Financial Engineering (MFE). This course is given in the second term (July 9-August 22) of the summer semester prior to the main academic year. The course is intended to be a stochastic boot camp, providing students a solid technical foundation on probability and stochastic processes in preparation for the main courses in financial engineering.

As a prerequisite, students are assumed to have already taken a semester course on probability theory and its applications, such as IEOR 3600, IEOR 4105 or SIEO 4150 at Columbia. Nevertheless, the first four classes are devoted to a review of basic probability theory, and much of the rest is intended to develop a solid foundation in probability theory.

In a first course on probability theory we learn about random variables and their probability distributions. In that first course, attention is usually focused on only one or two random variables. In this subsequent course we will extend the focus to stochastic processes, which are collections of random variables, usually indexed by time. (In stochastic process models, time can be regarded as either discrete or continuous.) For example, we might use stochastic processes to model the evolution of a stock price over time or the damage claims received by an insurance company over time, all of which evolve with considerable uncertainty.

Among the stochastic processes to be considered in this course are discrete-time Markov chains, random walks, martingales, continuous-time Markov chains, Poisson processes, birth-and-death processes, renewal processes, renewal-reward processes, Brownian motion and geometric Brownian motion.

The emphasis will be on stochastic processes and analysis techniques especially relevant for financial engineering. For example, among the topics to be covered are: the gambler's ruin problem, the optional stopping theorem, binomial lattice models, and the Black-Scholes formula.