This course is structured into three parts:
-- We begin with a general introduction to dynamic optimization. This introduction will allow us to formulate problems arising from applications in economics, engineering, and finance. We then introduce stochastic processes, Brownian motion, and Itô calculus. The connection between certain linear PDEs and diffusion processes will be demonstrated (Feynman–Kac representation, Dirichlet problem representation, Black–Scholes equations).
-- Next, we will explore the field of stochastic optimal control, highlighting the dynamic programming principle as a central pillar of our study. We will also introduce its infinitesimal counterpart, namely the Hamilton–Jacobi equation. Our attention will then turn to regular solutions of this equation, adopting a verification approach. These concepts will be illustrated through examples drawn from financial theory, thereby demonstrating their practical relevance. Furthermore, we will address the case of irregular value functions by introducing the theory of viscosity solutions. We will examine in detail the viscosity properties of the value function in a control problem, together with a uniqueness result by comparison.
-- Finally, we present numerical approximation methods based on finite differences for solving PDEs arising in finance. We will analyze the stability, consistency, and convergence properties of these schemes. Since the PDEs encountered in finance are often nonlinear, their discretization leads to semi-smooth systems, which will be solved using a nonsmooth Newton-type method (policy iteration). In addition, we will introduce reinforcement learning techniques for optimal stopping and optimal control problems, illustrating how methods such as Q-learning or policy gradient approaches can be used to price American options or to compute optimal investment strategies. The theoretical and numerical concepts will be validated through the computation of option prices (European and American put/call).
The evaluation will be based on projects carried out in pairs. Each pair will be assigned a topic related to a specific type of financial product. The numerical methods introduced during the course will need to be adapted to this product. At the end of the course, each pair will give an oral presentation of their project, explaining the financial model (e.g., Asian options, Merton’s portfolio, lookback options, jump–diffusion models, convertible bonds, etc.), presenting the numerical solution, and analyzing the simulation results.
- Enseignant: Forcadel Nicolas
- Enseignant: Zidani Housnaa