噪声可退化且依赖于状态和分布的平均场博弈

Mean-Field Game with Degenerate State-and Distribution-Dependent Noises

文章考虑状态方程关于状态和控制仿射,效用关于状态和控制凸的平均场博弈,允许状态方程的扩散项可退化且依赖状态和分布.由于允许漂移项和扩散项关于分布可以线性增长,因此可以包含线性二次平均场博弈,且允许状态的期望以线性形式出现在状态方程中.作者证明了对应的McKean-Vlasov型正倒向微分方程解的存在性,并获得了对应的解耦函数的正则性.最后作者证明了用平均场博弈的解和解耦函数可以以N-1/d+4的速度逼近多人博弈的Nash均衡.

The mean-field game is studied for state-affine systems with degenerate state-and distribution-dependent noises.The mean field terms of both drift and diffusion coefficients are allowed to grow in the distribution in a linear way,and therefore the linear-quadratic case(where the expected state appears in a linear way in the system dynamics)is included.The authors prove existence of the solution to the associated forward-backward stochastic differential equations of a McKean-Vlasov type and regularity of the decoupled function.Finally,they prove that solutions of the mean-field game together with the decoupled function approximate the Nash equilibrium of the N-players’game with an order up to N-1/d+4.