Optimal Liquidation in Target Zone Models and Neumann Problem of Backward SPDEs with Singular Terminal Condition

TL;DR

This paper investigates optimal liquidation strategies within target zone models, characterizing the value function through a Neumann problem of backward SPDEs with singular terminal conditions, and establishing existence, uniqueness, and links to stochastic differential equations.

Contribution

It introduces a novel approach to solve control problems with reflections using backward SPDEs with singular terminal conditions, proving existence and uniqueness of solutions.

Findings

Existence and uniqueness of strong solutions to the BSPDEs are established.

A comparison theorem for the BSPDEs is proved.

A new connection between forward-backward SDEs and BSPDEs is demonstrated.

Abstract

We study the optimal liquidation problems in target zone models using dynamic programming methods. Such control problems allow for stochastic differential equations with reflections and random coefficients. The value function is characterized with a Neumann problem of backward stochastic partial differential equations (BSPDEs) with singular terminal conditions. The existence and the uniqueness of strong solution to such BSPDEs are addressed, which in turn yields the optimal feedback control. In addition, the unique existence of strong solution to Neumann problem of general semilinear BSPDEs in finer functions space, a comparison theorem, and a new link between forward-backward stochastic differential equations and BSPDE are proved as well.