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Certain properties related to well posedness of switching diffusions

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  • Nguyen, Dang Hai
  • Yin, George
  • Zhu, Chao

Abstract

This work is devoted to switching diffusions that have two components (a continuous component and a discrete component). Different from the so-called Markovian switching diffusions, in the setup, the discrete component (the switching) depends on the continuous component (the diffusion process). The objective of this paper is to provide a number of properties related to the well posedness. First, the differentiability with respect to initial data of the continuous component is established. Then, further properties including uniform continuity with respect to initial data, and smoothness of certain functionals are obtained. Moreover, Feller property is obtained under only local Lipschitz continuity. Finally, an example of Lotka–Volterra model under regime switching is provided as an illustration.

Suggested Citation

  • Nguyen, Dang Hai & Yin, George & Zhu, Chao, 2017. "Certain properties related to well posedness of switching diffusions," Stochastic Processes and their Applications, Elsevier, vol. 127(10), pages 3135-3158.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:10:p:3135-3158
    DOI: 10.1016/j.spa.2017.02.004
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    References listed on IDEAS

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    1. Yuan, Chenggui & Mao, Xuerong, 2003. "Asymptotic stability in distribution of stochastic differential equations with Markovian switching," Stochastic Processes and their Applications, Elsevier, vol. 103(2), pages 277-291, February.
    2. Huang, Gang & Mandjes, Michel & Spreij, Peter, 2016. "Large deviations for Markov-modulated diffusion processes with rapid switching," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1785-1818.
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    Cited by:

    1. Xianggang Lu & Lin Sun, 2023. "Discounted Risk-Sensitive Optimal Control of Switching Diffusions: Viscosity Solution and Numerical Approximation," Mathematics, MDPI, vol. 12(1), pages 1-24, December.
    2. Li, Dagen & Liu, Meng, 2020. "Invariant measure of a stochastic food-limited population model with regime switching," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 16-26.
    3. Xinghu Jin & Tian Shen & Zhonggen Su & Yuzhen Tan, 2025. "The Euler-Maruyama Approximation of State-Dependent Regime Switching Diffusions," Journal of Theoretical Probability, Springer, vol. 38(1), pages 1-40, March.
    4. Liu, Meng & Bai, Chuanzhi, 2020. "Optimal harvesting of a stochastic mutualism model with regime-switching," Applied Mathematics and Computation, Elsevier, vol. 373(C).
    5. Hening, A. & Tran, K. Q. & Ungureanu, S., 2021. "The Effects of Random and Seasonal Environmental Fluctuations on Optimal Harvesting and Stocking," Working Papers 21/05, Department of Economics, City University London.
    6. Wang, Zhaojuan & Deng, Meiling & Liu, Meng, 2021. "Stationary distribution of a stochastic ratio-dependent predator-prey system with regime-switching," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).

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