Singular Perturbations of Nonlocal HJB Equations in Multiscale Stochastic Control

TL;DR

This paper studies multiscale stochastic control problems with nonlocal HJB equations driven by stable Lévy noises, deriving an effective averaged equation and proving convergence with explicit rates as the scale separation vanishes.

Contribution

It introduces a rigorous averaging principle for multiscale control systems with nonlocal dynamics and provides explicit convergence rates for the value functions.

Findings

Derived the limiting effective equation as the scale parameter tends to zero.

Proved convergence of the value functions to the solution of the averaged problem.

Established an explicit convergence rate for the approximation.

Abstract

This paper investigates a class of multiscale stochastic control problems driven by $\alpha$-stable L\'evy noises, where the controlled dynamics evolve across separate slow and fast time scales. The associated value functions are governed by a family of nonlocal Hamilton-Jacobi-Bellman (HJB) equations subject to singular perturbations. By employing the perturbed test function method, we carefully analyze this singular perturbation problem and derive a limiting effective equation as the time-scale separation parameter $\varepsilon$ approaches zero. This limiting equation characterizes the value function of the averaged control problem, thereby establishing a rigorous averaging principle for the original multiscale system. The effective Hamiltonian-along with the corresponding averaged control problem is obtained by averaging with respect to the invariant measure of the fast process. Moreover, we provide a probabilistic proof of convergence and establish an explicit convergence rate for the value functions.