STOCHASTIC FOKKER-PLANCK EQUATIONS 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 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. [ABSTRACT FROM AUTHOR]