On Maximum Principle of Non Linear Stochastic Mckean-Vlasov System with Applications Presented

Abstract

Partially observed optimal control problem has a variety of important applications in many fields and offers practical avenues for addressing real-world control challenges and decision-making problems, such as engineering, economics, and finance. The aim of this thesis is to study this kind of partially observed optimal control problem for forward-backward stochastic differential equations of the McKean– Vlasov type. The coefficients of the system and the cost functional depend on the state of the solution process as well as of its probability law and the control variable. We start by defining the primary tool (the partial derivative with respect to the probability measure in Wasserstein space) used to illustrate our main result. Then, we prove the necessary and sufficient conditions of optimality for FBSDEs of the McKean– Vlasov type under the assumption that the control domain is supposed to be convex. This result is based on Girsavov’s theorem. Finally, we prove a stochastic maximum principle for this kind of partially observed optimal control problems of McKean– Vlasov type driven by a Poisson random measure and an independent Brownian motion. As an illustration, a partially observed linear–quadratic control problem is studied in terms of stochastic filtering.