- 5MMCSAP

Goal(s)

Stochastic control is a classical topic in applied mathematics and arises in many practical situations where decisions must be made under uncertainty. In recent years, it has seen significant developments, primarily driven by problems in financial mathematics. To name just a few applications: option pricing and hedging, portfolio selection, risk management, real options and investment, optimal asset liquidation, and high-frequency trading.
The aim of this course is to provide an overview of the main methods and results in this field.

By the end of the course, students will be able to:

1) Understand key economic and financial decision-making problems under random uncertainty
2) Model market dynamics and propose optimal investment strategies
3) Understand, model, and manage financial risks in the simplest cases
4) Control financial risks through optimal portfolio allocation

Responsible(s)

Jérôme LELONG

Content(s)

We first present the standard approach of dynamic programming equations and the solution via verification, highlighting the limitations of this method. We then move on to the viscosity solutions approach: it requires more theory and technical tools but provides the general mathematical framework for dealing with stochastic control in a Markovian setting. We will then focus on another important class of stochastic control problems, namely optimal stopping, which commonly arises in the pricing of American options. The final part will be devoted to the martingale approach to portfolio optimization. The various methods presented in these lessons will be illustrated by several emerging applications in the fields of economics and finance.

The program is detailed as follows:

1)Introduction to the problem
2) Standard approach via dynamic programming and verification
3) Application examples: optimal portfolio allocation in the Merton case and under stochastic volatility
4)Optimal stopping problems: the case of American options
5)Limitations of the standard approach
6)The case of external constraints: example of optimal portfolio allocation under position constraints
7)Irreversible and reversible investment problems

Test

Evaluation : Examen écrit (2h)

Resit : Examen écrit (2h)

Calendar

The course exists in the following branches:

• Curriculum - Financial Engineering - Semester 9

Additional Information

Course ID : 5MMCSAP
Course language(s):

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