Singular perturbations in stochastic optimal control with unbounded data

TL;DR

This paper investigates singular perturbations in two-scale stochastic control systems with unbounded data, establishing convergence of the value function to an effective Hamilton-Jacobi-Bellman equation, with applications to neural network relaxation problems.

Contribution

It introduces a framework for analyzing singular perturbations in complex stochastic control systems with unbounded data, including the construction of effective Hamiltonians and convergence proofs.

Findings

Proved convergence of the value function to an effective HJB equation.

Constructed effective Hamiltonian and initial data for the limit problem.

Applied methods of probability, viscosity solutions, and homogenization.

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.