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Singular Perturbations of Nonlocal HJB Equations in Multiscale Stochastic Control

Qi Zhang, Yanjie Zhang, Ao Zhang

Abstract

This paper investigates a class of multiscale stochastic control problems driven by $α$-stable Lévy 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.

Singular Perturbations of Nonlocal HJB Equations in Multiscale Stochastic Control

Abstract

This paper investigates a class of multiscale stochastic control problems driven by -stable Lévy 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 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.

Paper Structure

This paper contains 10 sections, 15 theorems, 168 equations.

Table of Contents

  1. Introduction.
  2. Assumptions and the main results
  3. Convergence of nonlocal HJB equations
  4. Viscosity solutions
  5. Effective Hamiltonian and effective initial value
  6. The convergence result
  7. Probabilistic method
  8. Averaging principle of the controlled jump diffusion
  9. Proof of Theorem \ref{['intm']}
  10. Concluding remarks.

Key Result

Theorem 2.1

Under assumptions ($\mathbf{A_{b}}$), ($\mathbf{A_{c}}$), ($\mathbf{A_{g}}$), and ($\mathbf{A_{L}}$), the value function $u^{\varepsilon}$ of the multiscale stochastic control problem (mulSC) converges uniformly to the viscosity solution $\Bar{u}$ of the effective HJB equation 0EHJBu. That is, for e

Theorems & Definitions (31)

  • Remark 1
  • Remark 2
  • Theorem 2.1
  • Theorem 2.2
  • Remark 3
  • Definition 3.1
  • Remark 4
  • Definition 3.2
  • Corollary 3.1
  • Corollary 3.2
  • ...and 21 more