In this paper we study stochastic optimal control problems of fully coupled forward-backward stochastic differential equations (FBSDEs). The recursive cost functionals are defined by controlled fully coupled FBSDEs. We use a new method to prove that the value functions are deterministic, satisfy the dynamic programming principle, and are viscosity solutions to the associated generalized Hamilton--Jacobi--Bellman (HJB) equations. For this we generalize the notion of stochastic backward semigroup introduced by Peng Topics on Stochastic Analysis, Science Press, Beijing, 1997, pp. 85--138. We emphasize that when $\sigma$ depends on the second component of the solution $(Y, Z)$ of the BSDE it makes the stochastic control much more complicated and has as a consequence that the associated HJB equation is combined with an algebraic equation. We prove that the algebraic equation has a unique solution, and moreover, we also give the representation for this solution. On the other hand, we prove a new local existence and uniqueness result for fully coupled FBSDEs when the Lipschitz constant of $\sigma$ with respect to $z$ is sufficiently small. We also establish a generalized comparison theorem for such fully coupled FBSDEs.

We get a new type of controlled backward stochastic differential equations (BSDEs), namely, the BSDEs, coupled with value function. We prove the existence and the uniqueness theorem as well as a comparison theorem for such BSDEs coupled with value function by using the approximation method. We get the related dynamic programming principle (DPP) with the help of the stochastic backward semigroup which was introduced by Peng in 1997. By making use of a new, more direct approach, we prove that our nonlocal Hamilton-Jacobi-Bellman (HJB) equation has a unique viscosity solution in the space of continuous functions of at most polynomial growth. These results generalize the corresponding conclusions given by Buckdahn et al. (2009) in the case without control.

The purpose of this paper is to study two-person zero-sum stochastic differential games, in which one player is a major one and the other player is a group of $N$ minor agents which are collectively playing, are statistically identical, and have the same cost functional. The game is studied in a weak formulation; this means in particular that we can study it as a game of the type “feedback control against feedback control." The payoff/cost functional is defined through a controlled backward stochastic differential equation, for which the driving coefficient is assumed to satisfy strict concavity-convexity with respect to the control parameters. This ensures the existence of saddle point feedback controls for the game with $N$ minor agents. We study the limit behavior of these saddle point controls and of the associated Hamiltonian, and we characterize the limit of the saddle point controls as the unique saddle point control of the limit mean-field stochastic differential game.

In the present paper we investigate the problem of the existence of a value for differential games without Isaacs condition. For this we introduce a suitable concept of mixed strategies along a partition of the time interval, which are associated with classical nonanticipative strategies (with delay). Imposing on the underlying controls for both players a conditional independence property, we obtain the existence of the value in mixed strategies as the limit of the lower as well as of the upper value functions along a sequence of partitions which mesh tends to zero. Moreover, we characterize this value in mixed strategies as the unique viscosity solution of the corresponding Hamilton–Jacobi–Isaacs equation.

In Buckdahn, Djehiche, Li, and Peng (2009), the authors obtained mean-field Backward Stochastic Differential Equations (BSDEs) in a natural way as a limit of some highly dimensional system of forward and backward SDEs, corresponding to a great number of “particles” (or “agents”). The objective of the present paper is to deepen the investigation of such mean-field BSDEs by studying their stochastic maximum principle. This paper studies the stochastic maximum principle (SMP) for mean-field controls, which is different from the classical ones. This paper deduces an SMP in integral form, and it also gets, under additional assumptions, necessary conditions as well as sufficient conditions for the optimality of a control. As an application, this paper studies a linear quadratic stochastic control problem of mean-field type.