Singular perturbations in stochastic optimal control with unbounded data

¹ Department of Mathematics “T. Levi-Civita”, University of Padova, via Trieste, 63, 35121 Padova, Italy
² RWTH Aachen University, Institut für Mathematik, RTG Energy, Entropy, and Dissipative Dynamics, Templergraben 55 (111810), 52062 Aachen, Germany

^* Corresponding author: bardi@math.unipd.it

Received: 27 July 2022
Accepted: 23 March 2023

Abstract

We study singular perturbations of a class of two-scale stochastic control systems with unbounded data. The assumptions are designed to cover some relaxation problems for deep neural networks. We construct effective Hamiltonian and initial data and prove the convergence of the value function to the solution of a limit (effective) Cauchy problem for a parabolic equation of HJB type. We use methods of probability, viscosity solutions and homogenization.