## Stochastic Control and Portfolio Optimization – Summer Term 2020

On this webpage you find all information on my seminar on **Stochastic Control and Portfolio Optimization** in the summer term 2020 at TU Berlin.

### Background

**Stochastic control problems** are stochastic optimization problems in which the dynamics of a stochastic process can be influenced with the objective of optimizing a certain goal functional. A classical example arises in mathematical finance in the context of **optimal investment**: The trader chooses a trading strategy (the control) to influence the behavior of the wealth process (the controlled stochastic process) with the objective of maximizing expected utility of wealth at a future date (the goal functional).

In this seminar, we discuss both the **abstract theory** of stochastic control (dynamic programming, Hamilton-Jacobi-Bellman equations, classical verification, viscosity solutions) as well as applications to **portfolio optimization** problems (Merton problem, transaction cost problems).

### Course Organization

Due to the ongoing lockdown of the university, the seminar will be held **online**. Until further notice, we shall use the video conference tool Zoom. We have regular weekly meetings each tuesday at

**12:00h to 12:45h**: Group 1**13:00h to 13:45h**: Group 2

The **first meeting** is scheduled for **April 21**, in which we will discuss organizational matters and assign student presentations.

The seminar will be split in two parts. In the first half of the semester, we develop the fundamental theory by reading selected chapters of Nizar Touzi’s book on **Optimal Stochastic Control, Stochastic Target Problems, and Backward SDEs**. The book is avaible for download on **Nizar Touzi’s website**. The idea is to prepare about 5 to 10 pages each week, which will then be discussed jointly in class. With this, all participants of the seminar will develop the necessary background to prepare for the presentations. We will adhere to the following **schedule**:

**April 28**: Stochastic Differential Equations (p. 11-15) and Stochastic Control Problems in Standard Form (p. 27-30)-
**May 05**: Dynamic Programming Principle (p. 30-35) and Dynamic Programming Equation (p. 35-38) - Feel free to check out the following article which closes the gap in the proof of the Dynamic Programming Principle:
**A Pseudo-Markov Property for Controlled Diffusion Processes**(Claisse, Talay, Tan) **May 12**: Classical Verification (p. 55-58) and the Merton Problem of Optimal Investment (p. 58-60)**May 19**: Nonsmooth Value Functions (p. 41-42) and Viscosity Solutions (p. 69-74)**May 26**: Comparison and Uniqueness (p. 74-80)**June 02**: Viscosity Property of the Value Function (p. 89-95)

Starting June 09, the second half of the semester is reserved for student presentations on more **specialized topics**. Presentations will be held in groups of 3 students and presentations should not exceed 90 minutes (discussions included). The following topics will be covered:

**June 09**: Backward Stochastic Differential Equations and the Maximum Principle**June 16**: Optimal Investment in Price Impact Models**June 23**: Optimal Investment with Transaction Costs**June 30**: Optimal Investment in Semimartingale Markets**July 07**: The Stochastic Perron Method**July 14**: Optimal Investment for Private Investors**July 21**: Machine Learning Algorithms for Stochastic Control Problems

A short introduction to each topic and the topic assignment will take place in our first meeting on April 21.

While the literature on the topic is in English, the seminar will be held in **German**. You should be familiar with **Mathematical Finance I** and **Probability Theory II** to be able to follow the contents of the seminar. The seminar will not be graded. A written roundup of the seminar talks is not required.