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.