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

Abstract

We consider a mean-field control problem with càdlàg 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.

Self-exciting price impact via negative resilience in stochastic order books

“…In this stream of literature, strategies of infinite variation were first included by Lorenz and Schied (2013), where they allow for a non-martingale dynamics in the unaffected price, and hence the execution strategies need to account for the fluctuations in it. Recently, strategies of infinite variation emerge in related frameworks of Horst and Kivman (2021) and Fu et al (2022b). In the framework of Ackermann et al (2021a) we need to include strategies of infinite variation, as they actually come out as optimal trading schedules, e.g., to account for the fluctuations in (γ t ) and (ρ t ).…”