BSDEs with singular terminal condition and control problems with constraints

TL;DR

This paper introduces a probabilistic method using BSDEs with singular terminal conditions to solve non-Markovian control problems with state constraints, extending existing existence results.

Contribution

It presents a novel approach employing BSDEs with singular terminal conditions to address constrained control problems, generalizing prior existence theorems.

Findings

Existence of solutions to BSDEs with singular terminal conditions is established.

The approach applies to non-Markovian control problems with state constraints.

Explicit solutions are discussed for special cases.

Abstract

We provide a probabilistic solution of a not necessarily Markovian control problem with a state constraint by means of a Backward Stochastic Differential Equation (BSDE). The novelty of our solution approach is that the BSDE possesses a singular terminal condition. We prove that a solution of the BSDE exists, thus partly generalizing existence results obtained by Popier in [7] and [8]. We perform a verification and discuss special cases for which the control problem has explicit solutions.