，成果详细信息-凯发网

a classical problem in ergodic control theory consists in the study of the limit behaviour of λvλ(⋅) as λ↘0, when vλ 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 vλ 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 λvλ(⋅) is bounded and lipschitz uniformly with respect to λ>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 λvλ(.) and we characterize its limit w0 as maximal viscosity subsolution of a limit pde. using bsde methods we prove that w0 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 w0.