Backward stochastic differential equations with conditional reflection and related recursive optimal control problems

TL;DR

This paper introduces a new class of reflected backward stochastic differential equations (BSDEs) with reflection constraints based on conditional expectations, extending classical and mean reflection BSDEs, and explores their applications in optimal control and game theory.

Contribution

It defines and analyzes conditional RBSDEs, proving existence and uniqueness, and connects them to partial information optimal stopping and control problems, including new formulations of stochastic games.

Findings

Conditional RBSDEs generalize classical and mean reflection BSDEs.

Existence and uniqueness are established under mild conditions.

Applications to partial information optimal stopping, control, and zero-sum games are demonstrated.

Abstract

We introduce a new type of reflected backward stochastic differential equations (BSDEs) for which the reflection constraint is imposed on its main solution component, denoted as $Y$ by convention, but in terms of its conditional expectation $\mathbb{E}[Y_t|\mathcal{G}_{t}]$ on a general sub-filtration $\{\mathcal{G}_{t}\}.$ We thus term such equation as conditionally reflected BSDE (for short, conditional RBSDE). Conditional RBSDE subsumes classical RBSDE with a pointwise reflection barrier, and the recent developed BSDE with a mean reflection constraint, as its two special and extreme cases: they exactly correspond to $\{\mathcal{G}_{t}\}$ being the full filtration to represent complete information, and the degenerated filtration to deterministic scenario, respectively. For conditional RBSDE, we obtain its existence and uniqueness under mild conditions by combining the Snell envelope method with Skorokhod lemma. We also discuss its connection, in the case of linear driver, to a class of optimal stopping problems in presence of partial information. As a by-product, a new version of comparison theorem is obtained. With the help of this connection, we study weak formulations of a class of optimal control problems with reflected recursive functionals by characterizing the related optimal solution and value. Moreover, in the special case of recursive functionals being RBSDE with pointwise reflections, we study the strong formulations of related stochastic backward recursive control and zero-sum games, both in non-Markovian framework, that are of their own interests and have not been fully explored by existing literatures yet.