Ergodic BSDEs under weak dissipative assumptions

Abstract

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.

Approximation of the invariant law of SPDEs: error analysis using a Poisson equation for a full-discretization scheme

“…A proof of this result can be found in Section 6.1 in [12]. In this proof it is obvious that c and C are independent of h.…”

“…The idea of this decomposition is to apply the error estimates (12) and (15) for R h and P M respectively. We now use formula (20) onX h (t).…”

Representation of Homothetic Forward Performance Processes in Stochastic Factor Models Via Ergodic and Infinite Horizon BSDE

“…Indeed, since Z t , t ≥ 0, is bounded by C v /(C η − C v ) for both equations (15) and (42), the uniqueness can be proved along similar arguments used in Theorem 4.6 in [16] and Theorem 3.11 in [10].…”

“…In an infinite dimensional setting, an ergodic Lipschitz BSDE was introduced in [16] for the solution of an ergodic stochastic control problem; see also [8,10,36], and more recently [9] and [20] for various extensions. The infinite horizon quadratic BSDE was first solved in [6] by combining the techniques used in [7] and [22].…”

Ergodic BSDEs and related PDEs with Neumann boundary conditions under weak dissipative assumptions

“…To do that, it is easy to adapt the proof from [3] replacing the process U x t by its analogue in our context. Thus we define U x t as the strong solution of the following SDE:…”