Optimal control problem for reflected McKean-Vlasov SDEs

TL;DR

This paper studies the optimal control of reflected McKean-Vlasov stochastic differential equations, establishing the viscosity solution framework for associated Hamilton-Jacobi-Bellman equations on Wasserstein space, and proving uniqueness using a generalized distance function.

Contribution

It introduces a novel approach to characterize viscosity solutions of HJB equations on Wasserstein space for reflected McKean-Vlasov SDEs, including a new distance-like function for uniqueness.

Findings

Value function is a viscosity solution to HJB equations on Wasserstein space.

Dynamic programming principle holds for reflected McKean-Vlasov SDEs.

A new distance-like function ensures uniqueness of viscosity solutions.

Abstract

This work investigates the optimal control problem for reflected McKean-Vlasov SDEs and the viscosity solutions to Hamilton-Jacobi-Bellman(HJB) equations on the Wasserstein space in terms of intrinsic derivative. It follows from the flow property of reflected McKean-Vlasov SDEs that the dynamic programming principle holds. Applying the decoupling method and the heat kernel estimates for parabolic equations, we show that the value function is a viscosity solution to an appropriate HJB equation on the Wasserstein space, where the characterization of absolutely continuous curves on the Wasserstein space by the continuity equations plays an important role. To establish the uniqueness of viscosity solution, we generalize the construction of a distance like function initiated in Burzoni et al.(SICON, 2020) to the Wasserstein space over multidimensional space and show its effectiveness to cope with HJB equations in terms of intrinsic derivative on the Wasserstein space.