Forward-Backward Stochastic Linear-Quadratic Optimal Controls: Equilibrium Strategies and Non-Symmetric Riccati Equations
Qi Lü, Bowen Ma, Hanxiao Wang
TL;DR
This work tackles time-inconsistency in forward-backward stochastic linear-quadratic control by formulating a dynamic-game approach that yields time-consistent, closed-loop equilibrium strategies. The authors introduce an equilibrium Riccati equation (ERE), a coupled non-local non-symmetric system, and establish a priori bounds for the invariant $P_1(t,t)+P_2(t)^ op G_2(t)P_2(t)$ to prove existence and uniqueness of a solution and to construct the feedback law $\Theta(\cdot)$. They prove the key equivalence between the ERE and a forward–backward integral equation system (IES), from which the equilibrium value function $\mathbb{V}(t)=\tfrac{1}{2}\langle (P_1(t,t)+P_2(t)^ op G_2(t)P_2(t)) x, x \rangle$ is obtained. The results extend from smooth to non-smooth coefficients via mollification, providing a constructive method to obtain the equilibrium strategy with potential applications in finance and dynamic decision-making under time preference.
Abstract
Linear-quadratic optimal control problem for systems governed by forward-backward stochastic differential equations has been extensively studied over the past three decades. Recent research has revealed that for forward-backward control systems, the corresponding optimal control problem is inherently time-inconsistent. Consequently, the optimal controls derived in existing literature represent pre-committed solutions rather than dynamically consistent strategies. In this paper, we shift focus from pre-committed solutions to addressing the time-inconsistency issue directly, adopting a dynamic game-theoretic approach to derive equilibrium strategies. Owing to the forward-backward structure, the associated equilibrium Riccati equation (ERE) constitutes a coupled system of matrix-valued, non-local ordinary differential equations with a non-symmetric structure. This non-symmetry introduces fundamental challenges in establishing the solvability of the EREs. We overcome the difficulty by establishing a priori estimates for a combination of the solutions to EREs, which, interestingly, is a representation of the equilibrium value function.