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arXiv · 2410.22141

Singular Perturbations of Nonlocal HJB Equations in Multiscale Stochastic Control

Qi Zhang·Yanjie Zhang·Ao Zhang
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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.

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Qi Zhang, Yanjie Zhang, Ao Zhang. 2024-10-29. Singular Perturbations of Nonlocal HJB Equations in Multiscale Stochastic Control. https://arxiv.org/abs/2410.22141

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