A classical problem in ergodic control theory consists in the study of the limit behaviour of $\lambda V_\lambda(.)$ as $\lambda \to 0$, 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 $\lambda$. 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.

Based on a joint work with Rainer Buckdahn (Brest, France), Nana Zhao (Weihai, China).