A Stochastic Linear-Quadratic Leader-Follower Differential Game with Elephant Memory
Xinpo Li, Jingtao Shi
TL;DR
Addresses a stochastic linear-quadratic leader-follower differential game with elephant memory, where memory terms appear in both the state and diffusion components via $dx^{u^1,u^2}(t)$; the main approach derives an open-loop Stackelberg solution using two Riccati equations for the follower and leader and a matrix-valued L-equation, supported by an ABSDE/FBSDE framework based on Fréchet derivatives and dual operators. The follower and leader controls are obtained in state-feedback form through coupled forward–backward systems, with the follower's control given by $\bar{u}^1_t=-[\Xi^1(t)]^{-1}\{ d^1(t)^ op \Pi^1(t)c^2(t)x^{\bar{u}^1,u^2}_t + b^1(t)^ op \eta^1(t) + d^1(t)^ op \bar{\eta}^1(t) + d^1(t)^ op \Pi^1(t)d^2(t)u^2_t \}$ and the leader's by a memory-augmented feedback involving $\phi(t)=\xi(t)\bar{x}(t)$ and matrix functions $\Gamma(t),\Lambda(t,s)$. A practical dynamic advertising example demonstrates how elephant memory can shape hierarchical decisions, with Riccati solutions and auxiliary ABSDEs confirming the feasibility and interpretability of the results. Overall, the work extends stochastic Stackelberg control to memory-augmented dynamics and highlights both theoretical and applied implications for history-dependent strategic interactions.
Abstract
This paper is concerned with a stochastic linear-quadratic leader-follower differential game with elephant memory. The model is general in that the state equation for both the leader and the follower includes the elephant memory of the state and the control, which are part of the diffusion term. Under certain assumptions, the state feedback representation of the open-loop Stackelberg strategy is derived by introducing two Riccati equations and a special matrix-valued equation. Finally, theoretical results are illustrated by means of an example concerning a dynamic advertising problem with elephant memory.