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 reduced BSDE limit involving ergodic BSDEs and auxiliary control problems.

Contribution

It introduces a probabilistic approach to analyze the singular limit of BSDEs in two-scale stochastic systems in infinite dimensions, including the derivation of a reduced BSDE and its connection to ergodic BSDEs.

Findings

Value function converges to a reduced BSDE as scale ratio diverges.

Limit BSDE involves solutions of ergodic BSDEs.

Provides a probabilistic framework for two-scale stochastic control in infinite dimensions.

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 backward stochastic differential equation (BSDE) that it is shown to converge towards a 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 are required. The limit BSDE involves the solution of an ergodic BSDE and is itself interpreted as the value function of an auxiliary stochastic control problem on a reduced state space.