Ergodic BSDEs under weak dissipative assumptions

Arnaud Debussche (1, 2) , Ying Hu (2) , Gianmario Tessitore (3)

1 IPSO - Invariant Preserving SOlvers
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 Dipartimento di Matematica e Applicazioni [Milano]

In this paper we study ergodic backward stochastic differential equations (EBSDEs) dropping the strong dissipativity assumption needed in the previous work. 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.

Mots clés: Backward stochastic differential equation; Bismut-Elworthy formula; Coupling estimate; Ergodic control; Hamilton-Jacobi-Bellman equation; Recurrence property; Weak dissipative assumption

Domaines: Probabilités [math.PR]

Arnaud Debussche, Ying Hu, Gianmario Tessitore. Ergodic BSDEs under weak dissipative assumptions. Stochastic Processes and their Applications, 2011, 121 (3), pp.407-426. ⟨10.1016/j.spa.2010.11.009⟩. ⟨hal-00472146⟩