A state-constrained differential game arising in optimal portfolio liquidation

TL;DR

This paper studies a differential game model for multiple risk-averse agents competing in a market impact setting, establishing existence, uniqueness, and explicit solutions for Nash equilibria with state constraints.

Contribution

It introduces a novel linear-quadratic differential game framework with state constraints for portfolio liquidation, providing theoretical results and explicit solutions.

Findings

Existence and uniqueness of Nash equilibria proven.

Closed-form solutions derived in special cases.

Qualitative analysis of equilibrium strategies with financial insights.

Abstract

We consider $n$ risk-averse agents who compete for liquidity in an Almgren--Chriss market impact model. Mathematically, this situation can be described by a Nash equilibrium for a certain linear-quadratic differential game with state constraints. The state constraints enter the problem as terminal boundary conditions for finite and infinite time horizons. We prove existence and uniqueness of Nash equilibria and give closed-form solutions in some special cases. We also analyze qualitative properties of the equilibrium strategies and provide corresponding financial interpretations.