The course is an introduction to the basic theory of stochastic processes in discrete and continuous time (notably martingales, Markov processes, Brownian Motion, Ito Calculus, Girsanov's theorem, and optimal stopping, connections between PDEs and diffusions) and its basic applications in finance: pricing in complete and arbitrage free markets in discrete and continuous time, the Black-Scholes model).  

The textbook for the course is Introduction to Stochastic Calculus Applied to Finance, Second Edition, D. Lamberton and B. Lapeyre. We will cover the first 5 chapters of this book.