Long time asymptotics for fully nonlinear Bellman equations: a Backward SDE approach

Andrea Cosso (1) , Marco Fuhrman (2) , Huyen Pham (1, 3)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 Dipartimento di Matematica "F. Brioschi"
3 CREST - Centre de Recherche en Économie et Statistique

We study the large time behavior of solutions to fully nonlinear parabolic equations of Hamilton-Jacobi-Bellman type arising typically in stochastic control theory with control both on drift and diffusion coefficients. We prove that, as time horizon goes to infinity, the long run average solution is characterized by a nonlinear ergodic equation. Our results hold under dissipativity conditions, and without any nondegeneracy assumption on the diffusion term. Our approach uses mainly probabilistic arguments relying on new backward SDE representation for nonlinear parabolic, elliptic and ergodic equations.

Mots clés: ergodic type Bellman equation; Hamilton-Jacobi-Bellman type equation; stochastic control; large time behavior; backward SDE; ergodic type Bellman equation.

Domaines: Probabilités [math.PR]

Andrea Cosso, Marco Fuhrman, Huyen Pham. Long time asymptotics for fully nonlinear Bellman equations: a Backward SDE approach. Stochastic Processes and their Applications, 2016, 126 (7), pp.1932-1973. ⟨10.1016/j.spa.2015.12.009⟩. ⟨hal-01071396⟩