Obliquely Reflected BSDEs

TL;DR

This paper investigates multidimensional obliquely reflected backward stochastic differential equations, establishing existence and uniqueness results under weak assumptions, and applies these findings to randomized switching problems.

Contribution

It introduces new existence and uniqueness results for obliquely reflected BSDEs with flexible reflection directions in both Markovian and non-Markovian frameworks.

Findings

Existence of solutions under weak assumptions

Uniqueness results for non-Markovian cases

Application to randomized switching problems

Abstract

In this paper, we study existence and uniqueness to multidimensional Reflected Backward Stochastic Differential Equation in an open convex domain, allowing for oblique directions of reflection. In a Markovian framework, combining \emph{a priori} estimates for penalised equations and compactness arguments, we obtain existence results under quite weak assumptions on the driver of the BSDEs and the direction of reflection, which is allowed to depend on both $Y$ and $Z$. In a non Markovian framework, we obtain existence and uniqueness result for direction of reflection depending on time and $Y$. We make use in this case of stability estimates that require some smoothness condition on the domain and the direction of reflection. In a last Section, we illustrate the application of our theoretical results by introducing randomised switching problems.