Optimal control problem for reflected McKean--Vlasov stochastic differential equations with Poisson jumps

In this paper, we consider the optimal relaxed control problem for a class of one-dimensional reflected McKean--Vlasov stochastic differential equations with Poisson jumps.

Due to the presence of the jump term, the state process generally belongs to the Skorokhod space $D([0,T],\Rp)$, which makes the proof of tightness and the passage to the limit more complicated.

Under Lipschitz conditions and suitable growth conditions, we establish uniform moment estimates for the state process and the reflecting process.

Then, by using Aldous' tightness criterion, the continuity of the Skorokhod map, and the stability results for stochastic integrals, we prove the existence of an optimal relaxed control.

Furthermore, under the Roxin convexity condition, we prove the existence of a strict optimal control.

In the general case, we show that relaxed controls can be approximated by a sequence of strict controls.