Singular Limit of Two Scale Stochastic Optimal Control Problems in Infinite Dimensions by Vanishing Noise Regularization

TL;DR

This paper investigates the asymptotic behavior of value functions in two-scale stochastic control systems in infinite dimensions, using vanishing noise regularization to handle cylindrical noise and degeneracy issues.

Contribution

It introduces a novel vanishing noise regularization method to analyze the singular limit of value functions in complex infinite-dimensional stochastic control problems.

Findings

Limit of value function represented by a reduced control problem

Regularization technique overcomes cylindrical noise challenges

Provides insights into degenerate diffusion effects

Abstract

In this paper we study the limit of the value function for a two-scale, infinite-dimensional, stochastic controlled system with cylindrical noise and possibly degenerate diffusion. The limit is represented as the value function of a new reduced control problem (on a reduced state space). The presence of a cylindrical noise prevents representation of the limit by viscosity solutions of HJB equations, while degeneracy of diffusion coefficients prevents representation as a classical BSDE. We use a vanishing noise regularization technique.