Ergodic BSDEs with Multiplicative and Degenerate Noise

TL;DR

This paper investigates ergodic backward stochastic differential equations (BSDEs) in infinite dimensions with multiplicative, possibly degenerate noise, providing a stochastic representation for ergodic Hamilton-Jacobi-Bellman equations and applications to control problems.

Contribution

It introduces a new approach to ergodic BSDEs with degenerate noise using a concavity assumption, avoiding typical quantitative conditions and enabling applications to ergodic control of SPDEs.

Findings

Provides a stochastic representation for a class of ergodic HJB equations.

Enables synthesis of optimal feedback laws in ergodic control problems for SPDEs.

Handles degenerate noise under suitable conditions.

Abstract

In this paper we study an Ergodic Markovian BSDE involving a forward process $X$ that solves an infinite dimensional forward stochastic evolution equation with multiplicative and possibly degenerate diffusion coefficient. A concavity assumption on the driver allows us to avoid the typical quantitative conditions relating the dissipativity of the forward equation and the Lipschitz constant of the driver. Although the degeneracy of the noise has to be of a suitable type we can give a stochastic representation of a large class of Ergodic HJB equations; morever our general results can be applied to get the synthesis of the optimal feedback law in relevant examples of ergodic control problems for SPDEs.