Long time asymptotics for fully nonlinear Bellman equations: A backward SDE approach

Abstract

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.MSC Classification: 60H30, 60J60, 49J20.

Ergodic control of infinite-dimensional stochastic differential equations with degenerate noise

“…We end Section 5 proving that, under suitable assumptions, λ coincides with the value function of an ergodic control problem. This latter result is again proved using only probabilistic techniques, while in [5] the proof is based on PDE arguments (see Remark 6.2 for more details on this point).…”

“…To our knowledge, the only paper that deals, by means of BSDEs, with ergodic limits in the degenerate case is [5] where authors use the same tool of randomized control problems and related constrained BSDEs that we will eventually employ here. Notice however that in [5] the state process lives in a finite-dimensional Euclidean space and probabilistic methods are combined with PDE techniques, relying on powerful tools from the theory of viscosity solutions. Here, as already mentioned, we have to completely avoid these arguments.…”

“…As a matter of fact viscosity solutions require, in the infinite dimensional case, additional artificial assumptions that we can not impose here (see for instance the theory of B-continuous viscosity solutions for second order PDEs, [8], [19]). On the other side to separate difficulties we consider, as in [9] but differently from [5], only additive and uncontrolled noise. This in particular considerably simplifies the proof of estimate (4.3).…”

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

“…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].…”

Optimal switching problems with an infinite set of modes: An approach by randomization and constrained backward SDEs

“…A variant of the optimal switching problem is obtained by allowing for a switching at the terminal time, that is by removing the requirement that τ n = T and modifying the functional J 2 , introduced in (2.9), in the following way: 12) in order to take into account the cost of a switching at the final time. In some papers, the following condition is imposed on the data: for every a ∈ A and x ∈ C n ,…”