# Controlled Reflected SDEs and Neumann Problem for Backward SPDEs

## TL;DR

This paper addresses the solution of a control problem involving reflected stochastic differential equations with path-dependent coefficients, characterizing the value function via a Neumann boundary BSPDE and establishing existence and uniqueness results.

## Contribution

It introduces a novel approach to solving reflected SDE control problems with path dependence and proves general existence and uniqueness results for nonlinear BSPDEs with Neumann boundary conditions.

## Key findings

- Established existence and uniqueness of solutions for the Neumann BSPDEs.
- Constructed optimal feedback controls from the BSPDE solutions.
- Extended results to general nonlinear BSPDEs beyond the specific control problem.

## Abstract

We solve the optimal control problem of a one-dimensional reflected stochastic differential equation, whose coefficients can be path dependent. The value function of this problem is characterized by a backward stochastic partial differential equation (BSPDE) with Neumann boundary conditions. We prove the existence and uniqueness of sufficiently regular solution for this BSPDE, which is then used to construct the optimal feedback control. In fact we prove a more general result: The existence and uniqueness of strong solution for the Neumann problem for general nonlinear BSPDEs, which might be of interest even out of the current context.