Optimal Switching of One-Dimensional Reflected BSDEs, and Associated Multi-Dimensional BSDEs with Oblique Reflection

TL;DR

This paper studies optimal switching problems for one-dimensional reflected backward stochastic differential equations (RBSDEs) with costs, characterizing the value via multi-dimensional RBSDEs with oblique reflection, and explores existence, uniqueness, and applications.

Contribution

It introduces a novel framework linking optimal switching in RBSDEs to multi-dimensional obliquely reflected RBSDEs, with new existence and uniqueness results.

Findings

Existence of solutions shown via Picard iteration and penalization methods.

Uniqueness established through representation as value processes or stochastic differential games.

Interpretation of switched RBSDEs as real options in financial contexts.

Abstract

In this paper, an optimal switching problem is proposed for one-dimensional reflected backward stochastic differential equations (RBSDEs, for short) where the generators, the terminal values and the barriers are all switched with positive costs. The value process is characterized by a system of multi-dimensional RBSDEs with oblique reflection, whose existence and uniqueness are by no means trivial and are therefore carefully examined. Existence is shown using both methods of the Picard iteration and penalization, but under some different conditions. Uniqueness is proved by representation either as the value process to our optimal switching problem for one-dimensional RBSDEs, or as the equilibrium value process to a stochastic differential game of switching and stopping. Finally, the switched RBSDE is interpreted as a real option.