In this paper we introduce a new kind of Backward Stochastic Differential Equations, called ergodic BSDEs, which arise naturally in the study of optimal ergodic control. We study the existence, uniqueness and regularity of solution to ergodic BSDEs. Then we apply these results to the optimal ergodic control of a Banach valued stochastic state equation. We also establish the link between the ergodic BSDEs and the associated Hamilton-JacobiBellman equation. Applications are given to ergodic control of stochastic partial differential equations.

In this paper we study ergodic backward stochastic differential equations (EBSDEs) dropping the strong dissipativity assumption needed in [12]. In other words we do not need to require the uniform exponential decay of the difference of two solutions of the underlying forward equation, which, on the contrary, is assumed to be non degenerate.We show existence of solutions by use of coupling estimates for a non-degenerate forward stochastic differential equations with bounded measurable non-linearity. Moreover we prove uniqueness of "Markovian" solutions exploiting the recurrence of the same class of forward equations.Applications are then given to the optimal ergodic control of stochastic partial differential equations and to the associated ergodic Hamilton-Jacobi-Bellman equations.

We consider an Ito stochastic differential equation with delay, driven by brownian motion, whose solution, by an appropriate reformulation, defines a Markov process X with values in a space of continuous functions C, with generator L. We then consider a backward stochastic differential equation depending on X, with unknown processes (Y, Z), and we study properties of the resulting system, in particular we identify the process Z as a deterministic functional of X. We next prove that the forward-backward system provides a suitable solution to a class of parabolic partial differential equations on the space C driven by L, and we apply this result to prove a characterization of the fair price and the hedging strategy for a financial market with memory effects. We also include applications to optimal stochastic control of differential equation with delay: in particular we characterize optimal controls as feedback laws in terms the process X.

Abstract.In this note, we give the stochastic maximum principle for optimal control of stochastic PDEs in the general case (when the control domain need not be convex and the diffusion coefficient can contain a control variable).

We study a forward-backward system of stochastic differential equations in an infinite dimensional framekork and its relationships with a semilinear parabolic differential equation on a Hilbert space, in the spirit of the approach of Pardoux-Peng. We prove that the stochastic system allows to construct a unique solution of the parabolic equation in a suitable class of locally Lipschitz real functions. The parabolic equation is understood in a mild sense which requires the notion of a generalized directional gradient, that we introduce by a probabilistic approach and prove to exist for locally Lipschitz functions. The use of the generalized directional gradient allows to cover various applications to option pricing problems and to optimal stochastic control problems (including control of delay equations and reaction-diffusion equations), where the lack of differentiability of the coefficients precludes differentiability of solutions to the associated parabolic equations of Black-Scholes or Hamilton-Jacobi-Bellman type.