Multi-dimensional BSDE with Oblique Reflection and Optimal Switching

TL;DR

This paper investigates multi-dimensional BSDEs with oblique reflection, providing existence, uniqueness, and applications to optimal switching problems and variational inequalities, advancing the mathematical understanding of complex stochastic control systems.

Contribution

It introduces a novel approach to solving multi-dimensional BSDEs with oblique reflection and applies these results to optimal switching and variational inequalities.

Findings

Existence of solutions via penalization and convergence methods.

Uniqueness established through a verification approach.

Application to optimal switching problems and probabilistic interpretation of viscosity solutions.

Abstract

In this paper, we study a multi-dimensional backward stochastic differential equation (BSDE) with oblique reflection, which is a BSDE reflected on the boundary of a special unbounded convex domain along an oblique direction, and which arises naturally in the study of optimal switching problem. The existence of the adapted solution is obtained by the penalization method, the monotone convergence, and the a priori estimations. The uniqueness is obtained by a verification method (the first component of any adapted solution is shown to be the vector value of a switching problem for BSDEs). As applications, we apply the above results to solve the optimal switching problem for stochastic differential equations of functional type, and we give also a probabilistic interpretation of the viscosity solution to a system of variational inequalities.