A Constrained Control Problem with Degenerate Coefficients and Degenerate Backward SPDEs with Singular Terminal Condition

TL;DR

This paper investigates a constrained optimal control problem with degenerate coefficients, modeling portfolio liquidation with market impact, and establishes existence and uniqueness of solutions for the associated degenerate backward SPDE with singular terminal conditions.

Contribution

It introduces a novel approach to prove existence and uniqueness of solutions for degenerate BSPDEs with singular terminal conditions, relevant to financial models.

Findings

Proved existence and uniqueness of nonnegative solutions for degenerate BSPDEs.

Developed a new gradient estimate technique for degenerate BSPDEs.

Applied results to models of optimal portfolio liquidation.

Abstract

We study a constrained optimal control problem with possibly degenerate coefficients arising in models of optimal portfolio liquidation under market impact. The coefficients can be random in which case the value function is described by a degenerate backward stochastic partial differential equation (BSPDE) with singular terminal condition. For this degenerate BSPDE, we prove existence and uniqueness of a nonnegative solution. Our existence result requires a novel gradient estimate for degenerate BSPDEs.