Representation of limit values for nonexpansive stochastic differential games

Abstract A classical problem in ergodic control theory consists in the study of the limit behaviour of λ V λ ( ⋅ ) as λ ↘ 0 , when V λ 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 λ 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 λ V λ ( ⋅ ) 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 λ V λ ( . ) 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 .