Stochastic Optimal Control of Decoupled Reflected FBSDEs: A Variational Approach with Residual Reflection Terms
Abstract
This paper studies necessary optimality conditions for stochastic control problems governed by decoupled forward--backward stochastic differential equations with reflection constraints on the backward component. The cost functional is defined through the initial value of a reflected BSDE whose lower obstacle depends on the forward state. The nonsmooth nature of the reflection term prevents the direct application of classical variational methods. Using a penalization approach, spike variations, and duality arguments, we derive a penalized variational relation. A key feature of our method is to isolate the reflection contribution into a residual term, avoiding the differentiation of the nonsmooth penalization operator while preserving a Hamiltonian adjoint structure analogous to the classical stochastic maximum principle. Passing to the limit yields a reflected variational inequality of Pontryagin type, where the obstacle effect is explicitly represented through a residual contribution associated with the reflection term.
Related
- Reflected BSDEs under random horizons: Sharp support criteria, irregular reflection, and Cox reconstruction
- Averaged proximal reflected gradient method for monotone variational inequalities
- A Parameter-Free Adaptive Reflected Gradient Method for Monotone Variational Inequalities
- Conditional Reflected Backward Stochastic Differential Equations
- Nonsmooth Obstacles and Killed Resolvents in Reflected Stochastic Control
BibTeX
@misc{bengherbal2026stochastic,
title = {{Stochastic Optimal Control of Decoupled Reflected FBSDEs: A Variational Approach with Residual Reflection Terms}},
author = {Hanane Ben-Gherbal and Omar Kebiri},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09867},
eprint = {2610.09867},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09867},
}
RIS
TY - UNPB
TI - Stochastic Optimal Control of Decoupled Reflected FBSDEs: A Variational Approach with Residual Reflection Terms
AU - Hanane Ben-Gherbal
AU - Omar Kebiri
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09867
UR - https://arxiv.org/abs/2610.09867
AB - This paper studies necessary optimality conditions for stochastic control problems governed by decoupled forward--backward stochastic differential equations with reflection constraints on the backward component. The cost functional is defined through the initial value of a reflected BSDE whose lower obstacle depends on the forward state. The nonsmooth nature of the reflection term prevents the direct application of classical variational methods. Using a penalization approach, spike variations, and duality arguments, we derive a penalized variational relation. A key feature of our method is to isolate the reflection contribution into a residual term, avoiding the differentiation of the nonsmooth penalization operator while preserving a Hamiltonian adjoint structure analogous to the classical stochastic maximum principle. Passing to the limit yields a reflected variational inequality of Pontryagin type, where the obstacle effect is explicitly represented through a residual contribution associated with the reflection term.
ER -