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A Continuous-Time Ehrenfest Model with Catastrophes and Its Jump-Diffusion Approximation

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Abstract

We consider a continuous-time Ehrenfest model defined over the integers from \(-N\) to N, and subject to catastrophes occurring at constant rate. The effect of each catastrophe istantaneously resets the process to state 0. We investigate both the transient and steady-state probabilities of the above model. Further, the first passage time through state 0 is discussed. We perform a jump-diffusion approximation of the above model, which leads to the Ornstein-Uhlenbeck process with catastrophes. The underlying jump-diffusion process is finally studied, with special attention to the symmetric case arising when the Ehrenfest model has equal upward and downward transition rates.

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Acknowledgments

One of the authors (Selvamuthu Dharmaraja) thanks the National Board for Higher Mathematics, India, for the financial assistance during the preparation of this paper. The research of the remaining authors (Antonio Di Crescenzo, Virginia Giorno, Amelia G. Nobile) is partially supported by GNCS-INdAM and Regione Campania (legge 5).

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, 110016, India

    Selvamuthu Dharmaraja

  2. Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II n. 132, 84084, Fisciano, SA, Italy

    Antonio Di Crescenzo

  3. Dipartimento di Studi e Ricerche Aziendali (Management & Information Technology), Università di Salerno, Via Giovanni Paolo II n. 132, 84084, Fisciano (SA), Italy

    Virginia Giorno & Amelia G. Nobile

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  1. Selvamuthu Dharmaraja
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  2. Antonio Di Crescenzo
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  3. Virginia Giorno
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  4. Amelia G. Nobile
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Correspondence to Antonio Di Crescenzo.

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Dharmaraja, S., Di Crescenzo, A., Giorno, V. et al. A Continuous-Time Ehrenfest Model with Catastrophes and Its Jump-Diffusion Approximation. J Stat Phys 161, 326–345 (2015). https://doi.org/10.1007/s10955-015-1336-4

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  • DOI: https://doi.org/10.1007/s10955-015-1336-4

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