Singular limit of BSDEs and Optimal control of two scale stochastic systems in infinite dimensional spaces

TL;DR

This paper investigates the convergence of value functions in two-scale infinite-dimensional stochastic control systems using BSDEs, revealing a limit described by a reduced BSDE involving ergodic components.

Contribution

It introduces a probabilistic approach to analyze the singular limit of BSDEs in infinite dimensions for two-scale stochastic control systems, connecting the limit to ergodic BSDEs.

Findings

Value functions converge to a reduced BSDE as the scale ratio diverges.

The limit involves an ergodic BSDE and an auxiliary control problem.

Additive noise assumptions and non-degeneracy conditions are key.

Abstract

In this paper we study by probabilistic techniques the convergence of the value function for a two-scale, infinite-dimensional, stochastic controlled system as the ratio between the two evolution speeds diverges. The value function is represented as the solution of a \textit{backward stochastic differential equation} (BSDE) that it is shown to converge towards a \textit{reduced} BSDE. The noise is assumed to be additive both in the slow and the fast equations for the state. Some non degeneracy condition on the slow equation is required. The limit BSDE involves the solution of an \textit{ergodic} BSDE and is itself interpreted as the value function of an auxiliary stochastic control problem on a reduced state space.