Stochastic Fokker-Planck PIDE for conditional McKean-Vlasov jump diffusions and applications to optimal control

Abstract

The purpose of this paper is to study optimal control of conditional McKean–Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean–Vlasov jump diffusions, for short). To this end, we first prove a stochastic Fokker–Planck equation for the conditional law of the solution of such equations. Combining this equation with the original state equation, we obtain a Markovian system for the state and its conditional law. Furthermore, we apply this to formulate a Hamilton–Jacobi–Bellman equation for the optimal control of conditional McKean–Vlasov jump diffusions. Then we study the situation when the law is absolutely continuous with respect to Lebesgue measure. In that case the Fokker–Planck equation reduces to a stochastic partial differential equation for the Radon–Nikodym derivative of the conditional law. Finally we apply these results to solve explicitly the linear-quadratic optimal control problem of conditional stochastic McKean–Vlasov jump diffusions, and optimal consumption from a cash flow.