Backward Stochastic Differential Equations with Conditional Reflection and Related Recursive Optimal Control Problems

SIAM Journal on Control and Optimization, Volume 62, Issue 5, Page 2557-2589, October 2024.
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 [math] by convention, but in terms of its conditional expectation [math] on a general subfiltration [math]. We thus term such a equation as conditionally reflected BSDE (for short, conditional RBSDE). Conditional RBSDE subsumes classical RBSDE with a pointwise reflection barrier and the recently developed BSDE with a mean reflection constraint as its two special and extreme cases: they exactly correspond to [math] 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 the Skorokhod lemma. We also discuss its connection, in the case of a linear driver, to a class of optimal stopping problems in the presence of partial information. As a by-product, a new version of the 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 a non-Markovian framework, that are of their own interests and have not been fully explored by existing literature yet.