Advanced Topics in Stochastics Calculus

The goal of the course is to deepen knowledge of stochastic calculus and stochastic analysis. In the first part the theory of stochastic integration will be extended from Brownian motion to continuous local martingales. Moreover, based on such an extension, we will turn to other important topics such as: generalized Girsanov theorem, local times and Tanaka's formula as an extension of Ito's formula, and mainly stochastic filtering problems. In the second part of the course optimal control problems will be addressed, both with complete and partial observation, initially in discrete time (Markov decision processes) with possible extensions to continuous time.

Detailed knowledge of stochastic calculus for continuous semimartingales. Theory of stochastic filtering in discrete and continuous time. Markov decision processes, including partial observation.