Stochastic Control Optimization

Basic problems, methods and results in the theory of optimization of stochastic dynamical systems will be presented. Both discrete-time and continuous-time models will be considered, over finite and infinite horizon. Continuous-time models will be mostly described by stochastic differential equations. The main approaches will be dynamic programming, the study of the Hamilton-Jacobi-Bellman equation (including cases of solutions with low regularity), backward stochastic differential equations, the stochastic maximum principle (in the sense of Pontryagin). A brief introduction to other optimization problems will be given, such as optimal stopping or impulse control, and applications to basic models will be presented, for instance optimal investment problems in Mathematical Finance or linear quadratic optimal control.

Students attending the course will become acquainted with various classes of control and optimization problems for stochastic systems (with discrete time, with continuous time and formulated by stochastic differential equations, on finite and infinite horizon). They will learn the basic methods to solve such problems: dynamic programming and Hamilton-Jacobi-Bellman equations, backward stochastic differential equations, the stochastic maximum principle. They will also see how important models can be analyzed, such as optimal investment problems in Mathematical Finance and linear quadratic problems.