Course description
This is a course on the mathematical analysis of stochastic processes. It uses some modern measure theoretic terminology but is not mathematically rigorous. We start with discrete probability and Markov Chains and then move to continuous time Brownian motion and diffusion processes. The bulk of the class is devoted to stochastic integration, the Ito calculus, and the relation between partial differential equations and diffusion processes. We discuss the derivation of diffusion models and approximations. Applications and examples are taken from finance and physical sciences. Some of the homework exercises will involve computation.
Prerequisites
A mastery of multivariate calculus, multivariate probability, and linear algebra.
Computing
Some of the assignments will require lightweight computing (largely Monte Carlo) in C++. Students need some exposure to programming, preferably in C or C++, but other language (Java, Python, Matlab, VBA, Fortran, etc.) should do. Students will need access to a C++ compiler and Microsoft Excel (used for visualyzation only). Students with Linux operating systems can access the g++ compiler through a terminal window. Students with Apple OSX may need to install the developer tools to have access to g++. Students with Microsoft operating systems can use the cygwin package (recommended) or one of the integrated environments (Borland C++ or the Microsoft Developer Studio, less recommended).