Stochastic Control Optimization
A.Y. 2024/2025
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 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.
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.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
1) Introduction of stochastic control problems by means of applications to economics, finance and engineering.
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, ergodic control, optimal stopping problems, optimal switching, impulse control.
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, 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 prerequisites are: some notions on stochastic processes (in particular martingales and Markov processes), stochastic integration for Brownian motion and related stochastic calculus, stochastic differential equations driven by Brownian motion.
Teaching methods
Classroom lectures. Attendance is not compulsory, but it is strongly recommended.
Teaching Resources
Textbook:
H. Pham. Continuous-time Stochastic Control and Optimization with Financial Applications. Springer, 2009.
Remark: this book only covers most of the program, other references will be given during the lectures when needed.
H. Pham. Continuous-time Stochastic Control and Optimization with Financial Applications. Springer, 2009.
Remark: this book only covers most of the program, other references will be given during the lectures when needed.
Assessment methods and Criteria
The final examination consists of an oral exam.
During the exam, students will be asked to illustrate results presented during the course, in order to evaluate her/his knowledge and understanding of topics as well as the ability to apply them.
The final mark is on a 30-point scale.
During the exam, students will be asked to illustrate results presented during the course, in order to evaluate her/his knowledge and understanding of topics as well as the ability to apply them.
The final mark is on a 30-point scale.
MAT/06 - PROBABILITY AND STATISTICS - University credits: 6
Practicals: 12 hours
Lessons: 35 hours
Lessons: 35 hours
Professors:
Campi Luciano, Fuhrman Marco Alessandro
Shifts:
Educational website(s)
Professor(s)
Reception:
Monday, 10:30 am - 1:30 pm (upon appointment, possibly suppressed for academic duties)
Department of Mathematics, via Saldini 50, office 1017.