Stochastic Control Representations for Penalized Backward Stochastic Differential Equations

TL;DR

This paper establishes that penalized backward stochastic differential equations (BSDEs) can be represented through optimal stopping and control frameworks, providing new insights into their convergence and applications.

Contribution

It introduces novel optimal stopping and control representations for penalized BSDEs, including stopping at Poisson times and applications to multidimensional and constrained reflected BSDEs.

Findings

Convergence rate of penalized BSDEs derived from optimal stopping representation.

Stochastic control representations developed for multidimensional reflected BSDEs.

Application of the framework to reflected BSDEs with hedging constraints.

Abstract

This paper shows that penalized backward stochastic differential equation (BSDE), which is often used to approximate and solve the corresponding reflected BSDE, admits both optimal stopping representation and optimal control representation. The new feature of the optimal stopping representation is that the player is allowed to stop at exogenous Poisson arrival times. The convergence rate of the penalized BSDE then follows from the optimal stopping representation. The paper then applies to two classes of equations, namely multidimensional reflected BSDE and reflected BSDE with a constraint on the hedging part, and gives stochastic control representations for their corresponding penalized equations.