Backward stochastic differential equations with central value reflection

In this study, we investigate the well-posedness of a backward stochastic differential equation with jumps and a central value reflection constraint. The reflection condition is imposed on the real-valued function obtained by solving the equation E((Yₜ-x)) = 0 at each time t ∈ [0,T]. The driver depends on the distribution of the solution process Y and follows a general quadratic-exponential structure. The terminal value is assumed to be bounded. Using a fixed-point argument and Bounded Mean Oscillation (BMO in short) martingale theory, we establish the existence and uniqueness of local solutions, which are then extended to construct a global solution over the entire time interval $ [0,T] $.