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

Abstract : We introduce a new type of reflected backward stochastic differential equation (RBSDE) for which the reflection constraint is imposed on its main solution component, denoted as Y by convention, but in terms of its conditional expectation E[Yt|Gt] on a general sub-filtration {Gt}. 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 {Gt} 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 literature yet.
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Preprints, Working Papers, ... (Preprint)
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https://hal.archives-ouvertes.fr/hal-03850449
Contributor : Ying Hu Connect in order to contact the contributor
Submitted on : Sunday, November 13, 2022 - 5:28:54 PM
Last modification on : Thursday, November 17, 2022 - 4:00:04 AM

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Conditional_RBSDE-20221109.pdf
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  • HAL Id : hal-03850449, version 1

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Ying Hu, Jianhui Huang, Wenqiang Li. Backward stochastic differential equations with conditional reflection and related recursive optimal control problems. 2022. ⟨hal-03850449⟩

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