A classical problem in ergodic control theory consists in the study of the limit behaviour of $\lambda V_{\lambda }(\cdot )$ as $\lambda \searrow 0$, when $V_{\lambda }$ is the value function of a deterministic or stochastic control problem with discounted cost functional with infinite time horizon and discount factor λ. We study this problem for the lower value function $V_{\lambda }$ of a stochastic differential game with recursive cost, i.e., the cost functional is defined through a backward stochastic differential equation with infinite time horizon. But unlike the ergodic control approach, we are interested in the case where the limit can be a function depending on the initial condition. For this we extend the so-called non-expansivity assumption from the case of control problems to that of stochastic differential games and we derive that $\lambda V_{\lambda }(\cdot )$ is bounded and Lipschitz uniformly with respect to $\lambda >0$. Using PDE methods and assuming radial monotonicity of the Hamiltonian of the associated Hamilton-Jacobi-Bellman-Isaacs equation we obtain the monotone convergence of $\lambda V_{\lambda }(.)$ and we characterize its limit $W_0$ as maximal viscosity subsolution of a limit PDE. Using BSDE methods we prove that $W_0$ satisfies a uniform dynamic programming principle involving the supremum and the infimum with respect to the time, and this is the key for an explicit representation formula for $W_0$.

遍历控制理论中的一个经典问题在于对极限运动的研究。 $\lambda V_{\lambda }（\cdot ）$ 如 $\lambda \searrow 0$， 什么时候 $V_{\lambda }$是具有无限时间范围和折扣因子λ的具有折扣成本函数的确定性或随机控制问题的价值函数。我们针对低值函数研究此问题$V_{\lambda }$具有递归成本的随机微分博弈模型，即成本函数是通过具有无限时间范围的后向随机微分方程定义的。但是，与遍历控制方法不同，我们对极限可以是取决于初始条件的函数的情况感兴趣。为此，我们将所谓的非膨胀假设从控制问题扩展到了随机微分博弈的假设，并得出了$\lambda V_{\lambda }（\cdot ）$ 界和Lipschitz就 $\lambda >0$。使用PDE方法并假设相关联的Hamilton-Jacobi-Bellman-Isaacs方程的哈密顿量的径向单调性，我们获得的单调收敛性。$\lambda V_{\lambda }（。）$ 我们描述了它的极限 ${\mathrm{w\; ^}}_0$作为极限PDE的最大粘度子溶液。使用BSDE方法，我们证明${\mathrm{w\; ^}}_0$ 满足关于时间的最高和最低的统一动态规划原理，这是明确表示公式的关键。 ${\mathrm{w\; ^}}_0$。