A Mean Field Game of Optimal Portfolio Liquidation

TL;DR

This paper develops a mathematical framework for modeling optimal portfolio liquidation using mean field games, addressing the challenges of singular terminal conditions and proving existence and uniqueness of solutions.

Contribution

It introduces a novel approach to solve MFGs with singular terminal conditions via FBSDEs, extending continuation methods for linear-quadratic cases.

Findings

Proves existence and uniqueness of solutions to the MFG with singular terminal conditions.

Shows the MFG can be approximated by a sequence of problems with finite terminal values.

Provides a characterization of the MFG solution in terms of FBSDEs with singular drivers.

Abstract

We consider a mean field game (MFG) of optimal portfolio liquidation under asymmetric information. We prove that the solution to the MFG can be characterized in terms of a FBSDE with possibly singular terminal condition on the backward component or, equivalently, in terms of a FBSDE with finite terminal value, yet singular driver. Extending the method of continuation to linear-quadratic FBSDE with singular driver we prove that the MFG has a unique solution. Our existence and uniqueness result allows to prove that the MFG with possibly singular terminal condition can be approximated by a sequence of MFGs with finite terminal values.