Advanced Topics in Probability Theory

A.Y. 2019/2020
Overall hours
Learning objectives
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 described by stochastic differential equations. The main approach 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.
Expected learning outcomes
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.
Course syllabus and organization

Single session

Lesson period
First semester
Course syllabus
1) Discrete-time stochastic optimal control.

Stochastic controlled dynamical systems, payoff functionals over finite or infinite horizon. Value function, dynamic programming and the Hamilton-Jacobi-Bellman (HJB) equation. Applications to reference models.

2) Optimal control of stochastic differential equations.

Controlled stochastic differential equations, payoff functionals over finite and infinite horizon. Value function and the dynamic programming principle. Hamilton-Jacobi-Bellman (HJB) equations of parabolic and elliptic type. Verification theorems for regular solutions of the HJB equation. Linear-quadratic stochastic optimal control. Introduction to generalized solutions to the HJB equation, in the viscosity sense. Application to optimal portfolio problems.

3) Backward stochastic differential equations.

Formulation, existence and uniqueness results. Probabilistic representation of solutions to partial differential equations of semilinear type and of the value function of an optimal control problem. Stochastic maximum principle in the sense of Pontryagin.

4) Overview on other problems and methods.

Under certain circumstances, various other topics may also be presented, for instance control with partial observation, ergodic control, optimal stopping problems, optimal switching, impulse control.
Prerequisites for admission
Students attending the course are expected to have a relatively advanced knowledge of probability theory (based on measure theory). Other prerequisite topics for the course are: some notions on stochastic processes (in particular martingales and Markov processes), stochastic integration with respect to Brownian motion and related stochastic calculus, stochastic differential equations driven by Brownian motion; on these topics only some remainders will be given during the lectures.
Teaching methods
Classroom lectures. Attendance is not compulsory, but it is strongly recommended.
Teaching Resources

H. Pham. Continuous-time Stochastic Control and Optimization with Financial Applications. Springer, 2009.

Remark: this textbook only covers a part of the programme of the course. Lecture notes written by the teacher will cover many other topics. They will be made freely available on the course website.
Assessment methods and Criteria
The final examination consists of an oral exam.

In the oral exam, the student will be required to illustrate results as well as case studies presented during the course, in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/06 - PROBABILITY AND STATISTICS - University credits: 6
Lessons: 42 hours
Monday, 10:30 am - 1:30 pm (upon appointment, possibly suppressed for academic duties)
Department of Mathematics, via Saldini 50, office 1017.