Optimal stopping of conditional McKean–Vlasov jump diffusions

The purpose of this paper is to study the optimal stopping problem of conditional McKean–Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean–Vlasov jump diffusions, for short). We obtain sufficient variational inequalities for a function to be the value function of such a problem and for a stopping time to be optimal.The key is that we combine the conditional McKean–Vlasov equation with the associated stochastic Fokker–Planck partial integro-differential equation for the conditional law of the state. This leads to a Markovian system which can be handled by using a version of a Dynkin formula.Our verification result is illustrated by finding the optimal time to sell in a market with common noise and jumps.