The course consists of three parts.

Part 1 (Basics):  Optimization (Background from multivariate calculus, unconstraint optimization: necessary and sufficient conditions, constraint optimization: Lagrange and KKT);  Probability background (Probability space, random variables, integration, limit theorems, measure change);  Basics for stochastic processes in discrete time (filtrations, conditional expectations, stochastic processes, sub- and super martingales)

Part 2 (Discrete time finance): Mathematical finance in discrete time (setup and self-financing trading strategies, absence of arbitrage and fundamental theorems of asset pricing,  risk neutral pricing);  Stochastic control in discrete time (Supermartingales, optional sampling theorem, optimal stopping and  American options, discrete time control and the dynamic programming (Bellmann principle))

Part 3 (Stochastic processes in continuous time and basic Ito calculus): Stochastic processes and Brownian motion (Basic notions, stopping times and optional sampling, Brownian motion and Poisson process); First and quadratic variation, pathwise Ito formula, properties of Ito integrals)