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Numerical approximations and convergence analysis of piecewise diffusion Markov processes, with application to glioma cell migration

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  • Buckwar, Evelyn
  • Meddah, Amira

Abstract

In this paper, we focus on numerical approximations of Piecewise Diffusion Markov Processes (PDifMPs), particularly when the explicit flow maps are unavailable. Our approach is based on the thinning method for modelling the jump mechanism and combines the Euler-Maruyama scheme to approximate the underlying flow dynamics. For the proposed approximation schemes, we study both the mean-square and weak convergence. Weak convergence of the algorithms is established by a martingale problem formulation. Moreover, we employ these results to simulate the migration patterns exhibited by moving glioma cells at the microscopic level. Further, we develop and implement a splitting method for this PDifMP model and employ both the Thinned Euler-Maruyama and the splitting scheme in our simulation example, allowing us to compare both methods.

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  • Buckwar, Evelyn & Meddah, Amira, 2025. "Numerical approximations and convergence analysis of piecewise diffusion Markov processes, with application to glioma cell migration," Applied Mathematics and Computation, Elsevier, vol. 491(C).
    • Handle: RePEc:eee:apmaco:v:491:y:2025:i:c:s0096300324006945
    DOI: 10.1016/j.amc.2024.129233
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    References listed on IDEAS

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    1. Bertazzi, Andrea & Bierkens, Joris & Dobson, Paul, 2022. "Approximations of Piecewise Deterministic Markov Processes and their convergence properties," Stochastic Processes and their Applications, Elsevier, vol. 154(C), pages 91-153.
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    5. Peter Kritzer & Gunther Leobacher & Michaela Szolgyenyi & Stefan Thonhauser, 2017. "Approximation methods for piecewise deterministic Markov processes and their costs," Papers 1712.09201, arXiv.org, revised Jan 2019.
    6. Peter Kritzer & Gunther Leobacher & Michaela Szölgyenyi & Stefan Thonhauser, 2019. "Approximation methods for piecewise deterministic Markov processes and their costs," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2019(4), pages 308-335, April.
    7. Bierkens, Joris & Bouchard-Côté, Alexandre & Doucet, Arnaud & Duncan, Andrew B. & Fearnhead, Paul & Lienart, Thibaut & Roberts, Gareth & Vollmer, Sebastian J., 2018. "Piecewise deterministic Markov processes for scalable Monte Carlo on restricted domains," Statistics & Probability Letters, Elsevier, vol. 136(C), pages 148-154.
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