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.

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 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.

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.

In this paper we consider a mean-field backward stochastic differential equation (BSDE) driven by a Brownian motion and an independent Poisson random measure. Translating the splitting method introduced by Buckdahn et al. (2014) to BSDEs, the existence and the uniqueness of the solution (Yt,ξ,Zt,ξ,Ht,ξ), (Yt,x,Pξ,Zt,x,Pξ,Ht,x,Pξ) of the split equations are proved. The first and the second order derivatives of the process (Yt,x,Pξ,Zt,x,Pξ,Ht,x,Pξ) with respect to x, the derivative of the process (Yt,x,Pξ,Zt,x,Pξ,Ht,x,Pξ) with respect to the measure Pξ, and the derivative of the process (∂μYt,x,Pξ(y),∂μZt,x,Pξ(y),∂μHt,x,Pξ(y)) with respect to y are studied under appropriate regularity assumptions on the coefficients, respectively. These derivatives turn out to be bounded and continuous in L2. The proof of the continuity of the second order derivatives is particularly involved and requires subtle estimates. This regularity ensures that the value function V(t,x,Pξ)≔Ytt,x,Pξ is regular and allows to show with the help of a new Itô formula that it is the unique classical solution of the related nonlocal quasi-linear integral-partial differential equation (PDE) of mean-field type.

In this paper, we consider a mean-field stochastic differential equation, also called the McKean–Vlasov equation, with initial data (t,x)∈[0,T]×Rd, whose coefficients depend on both the solution Xt,xs and its law. By considering square integrable random variables ξ as initial condition for this equation, we can easily show the flow property of the solution Xt,ξs of this new equation. Associating it with a process Xt,x,Pξs which coincides with Xt,ξs, when one substitutes ξ for x, but which has the advantage to depend on ξ only through its law Pξ, we characterize the function V(t,x,Pξ)=E[Φ(Xt,x,PξT,PXt,ξT)] under appropriate regularity conditions on the coefficients of the stochastic differential equation as the unique classical solution of a nonlocal partial differential equation of mean-field type, involving the first- and the second-order derivatives of V with respect to its space variable and the probability law. The proof bases heavily on a preliminary study of the first- and second-order derivatives of the solution of the mean-field stochastic differential equation with respect to the probability law and a corresponding Itô formula. In our approach, we use the notion of derivative with respect to a probability measure with finite second moment, introduced by Lions in [Cours au Collège de France: Théorie des jeu à champs moyens (2013)], and we extend it in a direct way to the second-order derivatives.

In this paper we study differential games without Isaacs condition. The objective is to investigate on one hand zero-sum games with asymmetric information on the pay-off, and on the other hand, for the case of symmetric information but now for a non-zero sum differential game, the existence of a Nash equilibrium pay-off. Our results extend those by Buckdahn, Cardaliaguet and Rainer [SIAM J. Control Optim. 43 (2004) 624–642], to the case without Isaacs condition. To overcome the absence of Isaacs condition, randomization of the non-anticipative strategies with delay of the both players are considered. They differ from those in Buckdahn, Quincampoix, Rainer and Xu [Int. J. Game Theory 45 (2016) 795–816]. Unlike in [Int. J. Game Theory 45 (2016) 795–816], our definition of NAD strategies for a game over the time interval [ t,T ] (0 ≤ t ≤ T) guarantees that a randomized strategy along a partition π of [ 0,T ] remains a randomized NAD strategy with respect to any finer partition π′ (π ⊂ π′). This allows to study the limit behavior of upper and lower value functions defined for games in which the both players use also different partitions.

This work is devoted to the study of stochastic differential equations (SDEs) whose diffusion coefficient $\sigma(s,X_{\cdot\wedge s})$ is Lipschitz continuous with respect to the path of the solution process $X$, while its drift coefficient $b(s,X_{\cdot\wedge s},Q_{X_s})$ is only measurable with respect to $X$ and depends continuously (in the sense of the 1-Wasserstein metric) on the law of the solution process. Embedded in a mean-field game, the weak existence for such SDEs with mean-field term was recently studied by Lacker [Stochastic Process. Appl., 125 (2015), pp. 2856--2894] and Carmona and Lacker [Ann. Appl. Probab., 25 (2015), pp. 1189--1231] under only sequential continuity of $b(s,X_{\cdot\wedge s},Q_{X_s})$ in $Q_{X}$ with respect to a weak topology, but for uniqueness, Carmona and Lacker supposed that $b$ is independent of the mean-field term. We prove the uniqueness in law for $b$ depending on the mean-field, and the proof of the existence of a weak solution, relatively short in comparison with Carmona and Lacker's work, is extended in section 5 of this paper to the study of 2-person zero-sum stochastic differential games described by doubly controlled coupled mean-field forward-backward SDEs with dynamics whose drift coefficient is only measurable with respect to the state process.

We establish a new type of backward stochastic differential equations (BSDEs) connected with stochastic differential games (SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, where the lower and the upper value functions are defined through this BSDE. The existence and the uniqueness theorem and comparison theorem are proved for such equations with the help of an iteration method. We also show that the lower and the upper value functions satisfy the dynamic programming principle. Moreover, we study the associated Hamilton-Jacobi-Bellman-Isaacs (HJB-Isaacs) equations, which are nonlocal, and strongly coupled with the lower and the upper value functions. Using a new method, we characterize the pair (W,U) consisting of the lower and the upper value functions as the unique viscosity solution of our nonlocal HJB-Isaacs equation. Furthermore, the game has a value under the Isaacs' condition.