A Mean-Field Control Problem of Optimal Portfolio Liquidation with Semimartingale Strategies

TL;DR

This paper formulates a mean-field control problem for optimal portfolio liquidation involving semimartingale strategies, revealing the structure of the value function and the nature of optimal trading strategies.

Contribution

It introduces a novel mean-field control framework with semimartingale strategies for portfolio liquidation, deriving a coupled Riccati system and analyzing the optimal strategy jumps.

Findings

Value function depends only on the law of the state process.

The value function is linear-quadratic with coefficients satisfying Riccati equations.

Optimal strategies involve jumps only at the beginning and end of trading.

Abstract

We consider a mean-field control problem with c\`adl\`ag semimartingale strategies arising in portfolio liquidation models with transient market impact and self-exciting order flow. We show that the value function depends on the state process only through its law, and that it is of linear-quadratic form and that its coefficients satisfy a coupled system of non-standard Riccati-type equations. The Riccati equations are obtained heuristically by passing to the continuous-time limit from a sequence of discrete-time models. A sophisticated transformation shows that the system can be brought into standard Riccati form from which we deduce the existence of a global solution. Our analysis shows that the optimal strategy jumps only at the beginning and the end of the trading period.