[Paper Review] Mean-Field Leader-Follower Games with Terminal State Constraint

Guanxing Fu, Ulrich Horst|arXiv (Cornell University)|Sep 12, 2018

Stochastic processes and financial applications15 references4 citations

TL;DR

This paper establishes existence and uniqueness of solutions for linear McKean-Vlasov forward-backward SDEs with terminal state constraints, using a novel two-step continuation method involving a linear ansatz $ R = AQ + H $. The key contribution is a stochastic maximum principle for leader-follower portfolio liquidation games with asymmetric information and expectations feedback, proven via Cesàro convergence of penalized solutions.

ABSTRACT

We analyze linear McKean-Vlasov forward-backward SDEs arising in leader-follower games with mean-field type control and terminal state constraints on the state process. We establish an existence and uniqueness of solutions result for such systems in time-weighted spaces as well as a {convergence} result of the solutions with respect to certain perturbations of the drivers of both the forward and the backward component. The general results are used to solve a novel single-player model of portfolio liquidation under market impact with expectations feedback as well as a novel Stackelberg game of optimal portfolio liquidation with asymmetrically informed players.

Motivation & Objective

• To address leader-follower games with mean-field control and terminal state constraints on the state process, particularly in portfolio liquidation under asymmetric information.
• To establish existence and uniqueness of solutions for linear McKean-Vlasov FBSDEs with singular terminal conditions on the forward process and unknown terminal values for the backward process.
• To develop a stochastic maximum principle for mean-field leader-follower games by analyzing convergence of penalized solutions under perturbations of the drivers.
• To provide a rigorous analytical framework for solving novel single-player and Stackelberg games in optimal portfolio liquidation with expectations feedback.
• To lay the foundation for future numerical methods by characterizing the limit behavior of solutions to penalized problems.

Proposed method

• Uses a linear ansatz $ R_t = A_t Q_t + H_t $ to decouple the forward and backward components, transforming the system into an exogenous BSDE with singular terminal condition for $ A $ and a BSDE with known asymptotic behavior for $ H $.
• Applies a nested continuation argument in time-weighted spaces to prove existence and uniqueness of solutions to the FBSDE system under boundedness assumptions on model parameters.
• Introduces a penalized version of the original FBSDE system to approximate solutions under terminal state constraints, with convergence proven in $ L^ u $-norm for $ 1 < \nu < 2 $.
• Employs a fixed-point argument in a suitable function space to handle the coupled nature of $ (Q, H, R) $, overcoming limitations of standard continuation methods for FBSDEs with unknown terminal values.
• Uses a Cesàro averaging scheme over sequences of penalized solutions to establish convergence of optimal strategies and value functions to the constrained solution.
• Applies Itô's formula and a priori estimates to derive stability and convergence results, leveraging conditional expectations and Malliavin calculus techniques.

Experimental results

Research questions

• RQ1 Can existence and uniqueness be established for linear McKean-Vlasov FBSDEs with terminal state constraints on the forward process and unknown terminal values for the backward process?
• RQ2 How can a stochastic maximum principle be derived for mean-field leader-follower games with terminal state constraints, particularly in portfolio liquidation with asymmetric information?
• RQ3 What is the convergence behavior of solutions to penalized FBSDEs with respect to perturbations in the drivers $ \overline{f} $ and $ \overline{g} $, and in which topology does it occur?
• RQ4 Can the optimal strategy and value function of a constrained portfolio liquidation problem be recovered as the limit of solutions to penalized problems?
• RQ5 What is the role of expectations feedback and private information in shaping equilibrium strategies in Stackelberg-type portfolio liquidation games?

Key findings

• Existence and uniqueness of solutions to the linear McKean-Vlasov FBSDE with terminal state constraint are established in time-weighted spaces under boundedness assumptions on the coefficients.
• The solution to the FBSDE converges in the $ L^ u $-norm for $ 1 < \nu < 2 $ with respect to perturbations in the drivers $ \overline{f} $ and $ \overline{g} $, a result not obtainable via standard $ L^2 $-convergence.
• The optimal strategy for the leader in a portfolio liquidation game with terminal state constraint is shown to be the Cesàro limit of optimal strategies from a sequence of penalized problems.
• The value function of the constrained problem is the limit of convex combinations of value functions from penalized problems, confirming consistency of the approximation scheme.
• A novel stochastic maximum principle is derived for mean-field leader-follower games by analyzing the limit of penalized solutions, enabling characterization of equilibrium strategies under asymmetric information.
• The convergence of the adjoint process and state trajectory to the constrained solution is established via a combination of a priori estimates, Fatou’s lemma, and dominated convergence.