  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.
