In this paper we are interested in a new type of {\it mean-field}, non-Markovian stochastic control problems with partial observations. More precisely, we assume that the coefficients of the controlled dynamics depend not only on the paths of the state, but also on the conditional law of the state, given the observation to date. Our problem is strongly motivated by the recent study of the mean field games and the related McKean-Vlasov stochastic control problem, but with added aspects of path-dependence and partial observation. We shall first investigate the well-posedness of the state-observation dynamics, with combined reference probability measure arguments in nonlinear filtering theory and the Schauder fixed point theorem. We then study the stochastic control problem with a partially observable system in which the conditional law appears nonlinearly in both the coefficients of the system and cost function. As a consequence the control problem is intrinsically "time-inconsistent", and we prove that the Pontryagin Stochastic Maximum Principle holds in this case and characterize the adjoint equations, which turn out to be a new form of mean-field type BSDEs.

In this paper we study the optimal control problem for a class of general mean-field stochastic differential equations, in which the coefficients depend, nonlinearly, on both the state process as well as of its law. In particular, we assume that the control set is a general open set that is not necessary convex, and the coefficients are only continuous on the control variable without any further regularity or convexity. We validate the approach of Peng (SIAM J Control Optim 2(4):966–979, 1990) by considering the second order variational equations and the corresponding second order adjoint process in this setting, and we extend the Stochastic Maximum Principle of Buckdahn et al. (Appl Math Optim 64(2):197–216, 2011) to this general case.

Recently Buckdahn et al. (Mean-field stochastic differential equations and associated PDEs, arXiv:1407.1215, 2014) studied a mean-field stochastic differential equation (SDE), whose coefficients depend on both the solution process and also its law, and whose solution process (Xt,x,Pξs,Xt,ξs=Xt,x,Pξs|x=ξ), s∈[t,T],(t,x)∈[0,T]×Rd,ξ∈L2(Ft,Rd), admits the flow property. This flow property is the key for the study of the associated nonlocal partial differential equation (PDE). In this work we extend these studies in a non-trivial manner to mean-field SDEs which, in addition to the driving Brownian motion, are governed by a compensated Poisson random measure. We show that under suitable regularity assumptions on the coefficients of the SDE, the solution Xt,x,Pξ is twice differentiable with respect to x and its law. We establish the associated nonlocal integral-PDE, and we show that V(t,x,Pξ)=E[Φ(Xt,x,PξT,PXt,ξT)] is the unique classical solution V:[0,T]×Rd×P2(Rd)→R of this nonlocal integral-PDE with terminal condition Φ.

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.

In this paper we study the optimal stochastic control problem for stochastic differential systems reflected in a domain. The cost functional is a recursive one, which is defined via generalized backward stochastic differential equations developed by Pardoux and Zhang [20]. The value function is shown to be the unique viscosity solution to the associated Hamilton-Jacobi-Bellman equation, which is a fully nonlinear parabolic partial differential equation with a nonlinear Neumann boundary condition. For this, we also prove some new estimates for stochastic differential systems reflected in a domain.