Stochastic Maximum Principle for Optimal Liquidation with Control-dependent Terminal Time

TL;DR

This paper develops a stochastic maximum principle for optimal stock liquidation problems where the stopping time depends on the control, providing new theoretical insights and examples that differ from standard approaches.

Contribution

It introduces a novel SMP for control-dependent stopping times, expanding the theoretical framework for optimal liquidation problems.

Findings

The new SMP differs significantly from the standard version.

An example demonstrates the new SMP's applicability where standard SMP fails.

The approach broadens the understanding of control problems with variable stopping times.

Abstract

In this paper we study a general optimal liquidation problem with a control-dependent stopping time which is the first time the stock holding becomes zero or a fixed terminal time, whichever comes first. We prove a stochastic maximum principle (SMP) which is markedly different in its Hamiltonian condition from that of the standard SMP with fixed terminal time. We present a simple example in which the optimal solution satisfies the SMP in this paper but fails the standard SMP in the literature.