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 we show that it is of linear-quadratic form and that its coefficients satisfy a coupled system of nonstandard 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. Funding: Financial support is through the National Natural Science Foundation of China [Grants 12101465 and 12101523], Hong Kong Research Grants Council (Early Career Scheme) [Grant 25215122], Hong Kong Polytechnic University [Internal Grant P0044694, Internal Grant P0045668, and Startup Grant P0035348], and the Hong Kong Research Centre for Quantitative Finance [Grant P0042708].