Week 1 Discrete time Gaussian Markov processes

These are lecture notes for the class Stochastic Calculus offered at the Courant Institute in the Fall Semester of 2012. It is a graduate level class. Students should have a solid background in probability and linear algebra. The topic selection is guided in part by the needs of our MS program in Mathematics in Finance. But it is not focused entirely on the Black Scholes theory of derivative pricing. I hope that the main ideas are easier to understand in general with a variety of applications and examples. I also hope that the class is useful to engineers, scientists, economists, and applied mathematicians outside the world of finance. The term stochastic calculus refers to a family of mathematical methods for studying dynamics with randomness. Stochastic by itself means random, and it implies dynamics, as in stochastic process. The term calculus by itself has two related meanings. One is a system of methods for calculating things, as in the calculus of pseudo-differential operators or the umbral calculus. The tools of stochastic calculus include the backward equations and forward equations, which allow us to calculate the time evolution of expected values and probability distributions for stochastic processes. In simple cases these are matrix equations. In more sophisticated cases they are partial differential equations of diffusion type. The other sense of calculus is the study of what happens when ∆t → 0. In this limit, finite differences go to derivatives and sums go to integrals. Calculus in this sense is short for differential calculus and integral calculus, which refers to the simple rules for calculating derivatives and integrals – the product rule, the fundamental theorem of calculus, and so on. The operations of calculus, integration and differentiation, are harder to justify than the operations of algebra. But the formulas often are simpler and more useful: integrals can be easier than sums.