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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.22141 |
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| _version_ | 1866909894455590912 |
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| author | Zhang, Qi Zhang, Yanjie Zhang, Ao |
| author_facet | Zhang, Qi Zhang, Yanjie Zhang, Ao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22141 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Singular Perturbations of Nonlocal HJB Equations in Multiscale Stochastic Control Zhang, Qi Zhang, Yanjie Zhang, Ao Optimization and Control 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. |
| title | Singular Perturbations of Nonlocal HJB Equations in Multiscale Stochastic Control |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2410.22141 |