BSDE formulation of combined regular and singular stochastic control problems

TL;DR

This paper introduces a BSDE-based framework for combined regular and singular stochastic control problems, enabling analysis beyond Markovian models and providing approximation methods for optimal control.

Contribution

It develops a BSDE formulation for combined control problems, extending analysis to path-dependent cases and offering approximation techniques for optimal solutions.

Findings

BSDE formulation captures combined control problems.

Extension beyond Markovian models to path-dependent cases.

Approximate solutions for optimal controls using standard BSDEs.

Abstract

In this paper we study a class of combined regular and singular stochastic control problems that can be expressed as constrained BSDEs. In the Markovian case, this reduces to a characterization through a PDE with gradient constraint. But the BSDE formulation makes it possible to move beyond Markovian models and consider path-dependent problems. We also provide an approximation of the original control problem with standard BSDEs that yield a characterization of approximately optimal values and controls.