A classical problem in stochastic ergodic control consists of studying the limit behavior of the optimal value of a discounted integral in infinite horizon (the so called Abel mean of an integral cost) as the discount factor $\lambda$ tends to zero or the value defined with a Cesàro mean of an integral cost when the horizon $T$ tends to $+ \infty$. We investigate the possible limits in the norm of uniform convergence topology of values defined through Abel means or Ceàro means when $ \lambda \to 0^+ $ and $T \to + \infty $, respectively. Here we give two types of new representation formulas for the accumulation points of the values when the averaging parameter converges. We show that there is only one possible accumulation point which is the same for Abel means or Cesàro means. The first type of representation formula is based on probability measures on the product of the state space and the control state space, which are limits of occupational measures. The second type of representation formula is based on measures which are the projection of invariant measure on the space of relaxed controls. We also give a result comparing the both sets of measures involved in both classes of representation formulas. An important consequence of the representation formulas is the existence of the limit value when one has the equicontinuity property of Abel or Cesàro mean values. This is the case, for example, for nonexpansive stochastic control systems. In the end some insightful examples are given which help to better understand the results.

In this paper we consider one dimensional general mean-field backward stochastic differential equations (BSDEs), i.e., the generator of our mean-field BSDEs depends not only on the solution but also on the law of the solution. We first give a totally new comparison theorem for such type of BSDEs under Lipschitz condition. Furthermore, we study the existence of the solution of such mean-field BSDEs when the coefficients are only continuous and with a linear growth.

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.

In ergodic stochastic problems the limit of the value function Vλ of the associated discounted cost functional with infinite time horizon is studied, when the discounted factor λ tends to zero. These problems have been well studied in the literature and the used assumptions guarantee that the value function λVλ converges uniformly to a constant as λ→0. The objective of this work consists in studying these problems under the assumption, namely, the nonexpansivity assumption, under which the limit function is not necessarily constant. Our discussion goes beyond the case of the stochastic control problem with infinite time horizon and discusses also Vλ given by a Hamilton–Jacobi–Bellman equation of second order which is not necessarily associated with a stochastic control problem. On the other hand, the stochastic control case generalizes considerably earlier works by considering cost functionals defined through a backward stochastic differential equation with infinite time horizon and we give an explicit representation formula for the limit of λVλ, as λ→0.

We mainly investigate the existence of the Nash equilibrium payoffs for non-zero-sum stochastic differential games without assuming Isaacs condition in this paper. Along the partition π of the time interval , we choose a suitable random non-anticipative strategy with delay to study our non-zero-sum stochastic differential game. We prove for the corresponding both zero-sum stochastic differential games without Isaacs condition the existence of the value functions. With the help of these value functions we give the characterization of the Nash equilibrium payoffs. This characterization allows to prove the existence of Nash equilibrium payoffs.