Martingales On A Euclidean Manifold With A Boundary And Reflected BSDES In Non-Convex Domains

Abstract

The purpose of this paper is twofold. First, we introduce the notion of a $\Gamma$-martingale on a Euclidean manifold with a boundary (i.e., the closure of an open connected domain in R d ), we provide its equivalent characterization through the $\Gamma$-convex functions, and we establish its connection with the reflected backward stochastic differential equations (BSDEs) in the associated domain. Second, we show how the tools of stochastic geometry can be used to develop a new method for proving existence and uniqueness of solutions to reflected BSDEs. We implement this method and obtain a well-posedness result for reflected BSDEs in any bounded, two-dimensional, simply-connected domain that is locally C2 -diffeomorphic to a convex set. This work extends the results of [6] and [16].