This is a course on stochastic processes intended for people who will apply these ideas to practical problems. It covers mathematical terminology used to describe stochastic processes, including filtrations and transition probabilities. It uses some measure theoretic terminology but is not mathematically rigorous. The emphasis is on analytical tools (forward, backward equations, etc.) and computational methods (difference equations, simuation, Monte Carlo) for studying specific processes.
We start with Brownian motion and diffusion processes described by their short time mean (drift) and variance (quadratic variation). These are related to backward and forward partial differential equations. We discuss how to develop a diffusion process to model a physical or financial process. The Ito calculus is developed and related to stochastic differential equations (SDEs), change of measure (Girsanov) and the Feynman Kac formula. Linear Gaussian processes are described in more detail, in continuous and discrete time, with applications to filtering (estimation from noisy measurements). Time permitting, we may describe discrete Markov chains and give some analogous theory for them.