Speaker : Huyên Pham (Université Denis Diderot Paris 7)

Title : Contrôle d’équation de McKean-Vlasov et applications

Abstract :

This talk is concerned with the optimal control of stochastic (also called conditional) McKean-Vlasov equation. Such problem is motivated basically from the asymptotic formulation of Pareto-optimality for a large population of players (financial agents, firms) in mean-field interaction under a common noise. We develop the dynamic programming approach in this context. We first show a dynamic programming principle for the value function in the Wasserstein space of probability measures, which is proved from a flow property of the conditional law of the controlled state process. Next by relying on the notion of differentiability with respect to probability measures, introduced by P.L. Lions, and Itô’s formula along a flow of conditional measures, we derive the Hamilton-Jacobi-Bellman equation, and prove the viscosity property together with a uniqueness result. In the last part, we shall focus on the important class of linear quadratic control of conditional McKean-Vlasov equation with random coefficients, and provide closed-form solutions in terms of a decoupled system of backward stochastic Riccati equations. We illustrate our results with some examples and explicit solutions arising from financial applications including optimal trading models with price impact, and investment model of energy transition.