  We study the problem of optimal stopping 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. To achieve this, we combine the state equation
for the conditional McKean-Vlasov equation with the associated stochastic
Fokker-Planck equation for the conditional law of the solution of the state.
This gives us a Markovian system which can be handled by using a version of the
Dynkin formula. We illustrate our result by solving explicitly two optimal
stopping problems for conditional McKean-Vlasov jump diffusions. More
specifically, we first find the optimal time to sell in a market with common
noise and jumps, and, next, we find the stopping time to quit a project whose
state is modelled by a jump diffusion, when the performance functional involves
the conditional mean of the state.
