B0320 - STOCHASTIC CALCULUS II

Learning outcomes

At the end of the course, the students know the fundamentals of stochastic calculus, stochastic differential equations and the links with the theory of elliptic/parabolic partial differential equations. They can independently conduct the study of applied disciplines that require the knowledge of advanced tools of stochastic calculus and solve specific problems possibly by means of probabilistic numerical methods.

Course contents

Part I - Introduction to optimal control theory

Introduction to optimal control and dynamic optimization: Motivations and examples. Formulation of optimal control problems in discrete time and continuous time. Basic ideas and results of the dynamic programming method in the deterministic setting.

Part II - Stochastic optimal control

Discrete time case: Bellman's optimality principle; Bellman's equation and optimality conditions; verification theorems and transversality conditions in problems with infinite horizon.

Continuous time case: Bellman's optimality principle; Hamilton-Jacobi-Bellman equations: classical and viscosity solutions; some regularity results for HJB equations; verification theorems and transversality conditions. Applications to portfolio optimization.

Part III - Optimal stopping and singular stochastic control

Dynamic to programming equation and verification theorems. Applications American options and to irreversible investment problems.

Readings/Bibliography

1. Notes of the teacher
2. H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Springer-Verlag (2009).
3. J. Yong, X.Y. Zhou, Stochastic controls: Hamiltonian systems and HJB equations, Springer (1999).

Office hours

See the website of Salvatore Federico