IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v119y2009i7p2198-2221.html

Asymptotic properties of jump-diffusion processes with state-dependent switching

Author

Listed:
  • Xi, Fubao

Abstract

This work is concerned with a class of jump-diffusion processes with state-dependent switching. First, the existence and uniqueness of the solution of a system of stochastic integro-differential equations are obtained with the aid of successive construction methods. Next, the non-explosiveness is proved by truncation arguments. Then, the Feller continuity is established by means of introducing some auxiliary processes and by making use of the Radon-Nikodym derivatives. Furthermore, the strong Feller continuity is proved by virtue of the relation between the transition probabilities of jump-diffusion processes and the corresponding diffusion processes. Finally, on the basis of the above results, the exponential ergodicity is obtained under the Foster-Lyapunov drift conditions. Some examples are provided for illustration.

Suggested Citation

  • Xi, Fubao, 2009. "Asymptotic properties of jump-diffusion processes with state-dependent switching," Stochastic Processes and their Applications, Elsevier, vol. 119(7), pages 2198-2221, July.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:7:p:2198-2221
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(08)00168-3
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Wee, In-Suk, 1999. "Stability for multidimensional jump-diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 80(2), pages 193-209, April.
    2. John Buffington & Robert J. Elliott, 2002. "American Options With Regime Switching," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(05), pages 497-514.
    3. 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.
    4. Wu, Liming, 2001. "Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems," Stochastic Processes and their Applications, Elsevier, vol. 91(2), pages 205-238, February.
    5. Xi, Fubao, 2004. "Stability of a random diffusion with nonlinear drift," Statistics & Probability Letters, Elsevier, vol. 68(3), pages 273-286, July.
    6. Khasminskii, R.Z. & Zhu, C. & Yin, G., 2007. "Stability of regime-switching diffusions," Stochastic Processes and their Applications, Elsevier, vol. 117(8), pages 1037-1051, August.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Xinghu Jin & Tian Shen & Zhonggen Su, 2023. "Using Stein’s Method to Analyze Euler–Maruyama Approximations of Regime-Switching Jump Diffusion Processes," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1797-1828, September.
    2. Kexin Chen & Hoi Ying Wong, 2024. "Duality in optimal consumption–investment problems with alternative data," Finance and Stochastics, Springer, vol. 28(3), pages 709-758, July.
    3. Min Qian & Fuxi Zhang, 2011. "Entropy Production Rate of the Coupled Diffusion Process," Journal of Theoretical Probability, Springer, vol. 24(3), pages 729-745, September.
    4. Sabbar, Yassine & Kiouach, Driss & Rajasekar, S.P. & El-idrissi, Salim El Azami, 2022. "The influence of quadratic Lévy noise on the dynamic of an SIC contagious illness model: New framework, critical comparison and an application to COVID-19 (SARS-CoV-2) case," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    5. Sabbar, Yassine & Din, Anwarud & Kiouach, Driss, 2023. "Influence of fractal–fractional differentiation and independent quadratic Lévy jumps on the dynamics of a general epidemic model with vaccination strategy," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    6. Xi, Fubao & Zhu, Chao, 2018. "On the martingale problem and Feller and strong Feller properties for weakly coupled Lévy type operators," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 4277-4308.
    7. 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.
    8. Xi, Fubao & Yin, George, 2013. "The strong Feller property of switching jump-diffusion processes," Statistics & Probability Letters, Elsevier, vol. 83(3), pages 761-767.
    9. Chen, Xingzhi & Xu, Xin & Tian, Baodan & Li, Dong & Yang, Dan, 2022. "Dynamics of a stochastic delayed chemostat model with nutrient storage and Lévy jumps," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    10. Peng Chen & Xinghu Jin & Tian Shen & Zhonggen Su, 2024. "Variable-Step Euler–Maruyama Approximations of Regime-Switching Jump Diffusion Processes," Journal of Theoretical Probability, Springer, vol. 37(2), pages 1597-1626, June.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Xi, Fubao & Yin, G., 2010. "Asymptotic properties of nonlinear autoregressive Markov processes with state-dependent switching," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1378-1389, July.
    2. Alessandro Ramponi, 2011. "Mixture Dynamics and Regime Switching Diffusions with Application to Option Pricing," Methodology and Computing in Applied Probability, Springer, vol. 13(2), pages 349-368, June.
    3. Leonardo Videla & Rolando Rebolledo, 2022. "Evolving Systems of Stochastic Differential Equations," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1662-1705, September.
    4. Caraballo, Tomás & Settati, Adel & Fatini, Mohamed El & Lahrouz, Aadil & Imlahi, Abdelouahid, 2019. "Global stability and positive recurrence of a stochastic SIS model with Lévy noise perturbation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 677-690.
    5. Feifei Bian & Wencai Zhao & Yi Song & Rong Yue, 2017. "Dynamical Analysis of a Class of Prey-Predator Model with Beddington-DeAngelis Functional Response, Stochastic Perturbation, and Impulsive Toxicant Input," Complexity, Hindawi, vol. 2017, pages 1-18, December.
    6. Zhenya Liu & Yuhao Mu, 2022. "Optimal Stopping Methods for Investment Decisions: A Literature Review," IJFS, MDPI, vol. 10(4), pages 1-23, October.
    7. Luis H. R. Alvarez E. & Wiljami Sillanpää, 2023. "Optimal stopping and impulse control in the presence of an anticipated regime switch," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 98(2), pages 205-230, October.
    8. Tong, Jinying & Zhang, Zhenzhong & Bao, Jianhai, 2013. "The stationary distribution of the facultative population model with a degenerate noise," Statistics & Probability Letters, Elsevier, vol. 83(2), pages 655-664.
    9. Aleksandar Mijatovi'c & Martijn Pistorius, 2009. "Exotic derivatives under stochastic volatility models with jumps," Papers 0912.2595, arXiv.org, revised Oct 2010.
    10. Rujivan, Sanae & Thamrongrat, Nopporn & Juntanon, Parun & Djehiche, Boualem, 2025. "Analytical computation of conditional moments in the extended Cox–Ingersoll–Ross process with regime switching: Hybrid PDE system solutions with financial applications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 229(C), pages 176-202.
    11. Zhang, Tian & Chen, Huabin, 2019. "The stability with a general decay of stochastic delay differential equations with Markovian switching," Applied Mathematics and Computation, Elsevier, vol. 359(C), pages 294-307.
    12. Amogh Deshpande, 2011. "A Simple Novel Approach to Valuing Risky Zero Coupon Bond in a Markov Regime Switching Economy," Methodology and Computing in Applied Probability, Springer, vol. 13(4), pages 783-800, December.
    13. Anatoliy Swishchuk & Aiden Huffman, 2020. "General Compound Hawkes Processes in Limit Order Books," Risks, MDPI, vol. 8(1), pages 1-25, March.
    14. Hainaut, Donatien, 2012. "Multidimensional Lee–Carter model with switching mortality processes," Insurance: Mathematics and Economics, Elsevier, vol. 50(2), pages 236-246.
    15. Li, Zhi & Zhang, Wei, 2017. "Stability in distribution of stochastic Volterra–Levin equations," Statistics & Probability Letters, Elsevier, vol. 122(C), pages 20-27.
    16. Siu, Tak Kuen & Yang, Hailiang & Lau, John W., 2008. "Pricing currency options under two-factor Markov-modulated stochastic volatility models," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 295-302, December.
    17. Robert Elliott & Carlton-James Osakwe, 2006. "Option Pricing for Pure Jump Processes with Markov Switching Compensators," Finance and Stochastics, Springer, vol. 10(2), pages 250-275, April.
    18. Dong-Mei Zhu & Jiejun Lu & Wai-Ki Ching & Tak-Kuen Siu, 2019. "Option Pricing Under a Stochastic Interest Rate and Volatility Model with Hidden Markovian Regime-Switching," Computational Economics, Springer;Society for Computational Economics, vol. 53(2), pages 555-586, February.
    19. Bao, Jianhai & Hou, Zhenting & Yuan, Chenggui, 2009. "Stability in distribution of neutral stochastic differential delay equations with Markovian switching," Statistics & Probability Letters, Elsevier, vol. 79(15), pages 1663-1673, August.
    20. Godin, Frédéric & Lai, Van Son & Trottier, Denis-Alexandre, 2019. "Option pricing under regime-switching models: Novel approaches removing path-dependence," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 130-142.

    More about this item

    Keywords

    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:119:y:2009:i:7:p:2198-2221. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.