Stochastic Fokker-Planck PIDE for Conditional McKean-Vlasov Jump Diffusions and Applications to Optimal Control

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 an Hamilton-Jacobi-Bellman (HJB) 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 (SPDE) for the Radon-Nikodym derivative of the conditional law. Finally we apply these results to solve explicitly the following problems: -Linear-quadratic optimal control of conditional stochastic McKean-Vlasov jump diffusions. -Optimal consumption from a cash flow modelled as a conditional stochastic McKean-Vlasov differential equation with jumps.