A Moving Boundary Problem for Brownian Particles with Singular Forward-Backward Interactions

Abstract: We introduce a system of Brownian particles, each absorbed upon hitting an associated moving boundary. The boundaries are determined by the conditional probabilities of the particles being absorbed before some final time horizon, given the current knowledge of the system. While the particles evolve forward in time, the conditional probabilities are computed backwards in time, leading to a specification of the particle system as a system of singular forward-backward SDEs coupled through hitting times. Its analysis leads to a novel type of tiered moving boundary problem. Each level of this PDE corresponds to a different configuration of unabsorbed particles, with the boundary and the boundary condition of a given level being determined by the solution of the preceding one. We establish classical well-posedness for this moving boundary problem and use its solution to solve the original forward-backward system and prove its uniqueness.