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

TL;DR

This paper introduces a new framework for martingales on Euclidean manifolds with boundaries and applies stochastic geometry to establish existence and uniqueness of solutions for reflected backward stochastic differential equations in non-convex domains.

Contribution

It defines $\Gamma$-martingales on manifolds with boundaries, links them to reflected BSDEs, and develops a stochastic geometry-based method for proving well-posedness in non-convex domains.

Findings

Defined $\Gamma$-martingales and characterized them via $\Gamma$-convex functions.

Established a new method for existence and uniqueness of reflected BSDEs.

Proved well-posedness in bounded, two-dimensional, simply-connected 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].