Linear Quadratic Leader-follower Stochastic Differential Games: Closed-Loop Solvability
Abstract
In this paper, a leader-follower stochastic differential game is studied for a linear stochastic differential equation with a quadratic cost functional. The coefficients in the state equation and the weighting matrices in the cost functionals are all deterministic. Closed-loop strategies are introduced, which require to be independent of initial states; and such a nature makes it very useful and convenient in applications. The follower first solves a stochastic linear quadratic optimal control problem, and his optimal closed-loop strategy is characterized by a Riccati equation, together with an adapted solution to a linear backward stochastic differential equation. Then the leader turns to solve a stochastic linear quadratic optimal control problem of a forward-backward stochastic differential equation, necessary conditions for the existence of optimal closed-loop strategies for the leader is given by the existence of a Riccati equation. Some examples are also given.
Cite
@article{arxiv.2107.05240,
title = {Linear Quadratic Leader-follower Stochastic Differential Games: Closed-Loop Solvability},
author = {Zixuan Li and Jingtao Shi},
journal= {arXiv preprint arXiv:2107.05240},
year = {2021}
}
Comments
36 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:2012.14599; text overlap with arXiv:1607.04526, arXiv:1401.4754 by other authors