Forward-Backward Stochastic Differential Equations Generated by Bernstein Diffusions

Abstract

In this article, we prove new results regarding hitherto unknown relations that exist between certain Bernstein diffusions on the one hand and processes that typically occur in forward-backward systems of stochastic differential equations on the other hand. More specifically, we consider Bernstein diffusions that can wander in bounded convex domains where d is arbitrary, and which are generated there by a forward-backward system of decoupled linear deterministic parabolic partial differential equations. This makes them reversible Itô diffusions under some conditions that pertain to their marginal distributions, which then allows us to construct D × d ×d2 -valued processes that are weak solutions to suitably defined forward -backward systems of coupled stochastic differential equations. Moreover, we also consider the converse problem, namely, that of knowing whether the first component of a weak solution to a given forward-backward system is a Bernstein diffusion in some sense, which we solve affirmatively in a specific case.