A Mean Field Game of Optimal Portfolio Liquidation.

Abstract

We consider a mean field game (MFG) of optimal portfolio liquida- tion under asymmetric information. In the first part we recall the link between optimal liquidation and a (F)BSDE with singular terminal value, when there is only one player ([1], [2] and [3]) Then we will explain how the solution to the MFG can be characterized in terms of a mean-field FBSDE with possibly singular terminal condition on the backward component or, equivalently, in terms of a mean-field FBSDE with finite terminal value, yet singular driver. Extending the method of continuation to linear-quadratic FBSDE with singular driver we prove that this FBSDE has a unique solution. This solution provides an optimal control for the MFG and we also obtain a ε-Nash equilibrium when the number of players is increasing. Finally 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. Here contrary to the \classical" case and surprisingly, the penalized scheme does not directly give the solution of the initial FBSDE.