Generalized Mean-Reflected McKean--Vlasov BSDEs: Well-Posedness and a Markovian PDE Representation

Abstract

This paper investigates a class of generalized mean-reflected McKean--Vlasov type backward stochastic differential equations (BSDEs). Our new framework combines a mean reflection constraint on the solution's expectation with a generalized integral with respect to a continuous non-decreasing process. We establish the existence and uniqueness of the solution. The uniqueness is derived via stability estimates, while the existence is proved by employing a penalization method combined with a smooth approximation of the obstacle. We further study a Markovian case, where the equation reduces to a nonlocal parabolic PDE with a mean constraint. We establish the well-posedness and the probabilistic representation of this problem, and we describe the regularity of the multiplier in terms of the regularity of the obstacle.