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Some Results on the Telegraph Process Confined by Two Non-Standard Boundaries

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Abstract

We analyze the one-dimensional telegraph random process confined by two boundaries, 0 and H > 0. The process experiences hard reflection at the boundaries (with random switching to full absorption). Namely, when the process hits the origin (the threshold H) it is either absorbed, with probability α, or reflected upwards (downwards), with probability 1 − α, for 0 < α < 1. We provide various results on the expected values of the renewal cycles and of the absorption time. The adopted approach is based on the analysis of the first-crossing times of a suitable compound Poisson process through linear boundaries. Our analysis includes also some comparisons between suitable stop** times of the considered telegraph process and of the corresponding diffusion process obtained under the classical Kac’s scaling conditions.

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Acknowledgments

This work is partially supported by the group GNCS of INdAM (Istituto Nazionale di Alta Matematica), and by MIUR - PRIN 2017, project ‘Stochastic Models for Complex Systems’, no. 2017JFFHSH. We thank an anonymous referee for useful comments that allowed us to improve the paper.

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Authors and Affiliations

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

    Antonio Di Crescenzo, Barbara Martinucci & Paola Paraggio

  2. Department of Mathematical Sciences, Binghamton University, Binghamton, NY, 13902-6000, USA

    Shelemyahu Zacks

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  1. Antonio Di Crescenzo
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  2. Barbara Martinucci
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  3. Paola Paraggio
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  4. Shelemyahu Zacks
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Correspondence to Barbara Martinucci.

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Di Crescenzo, A., Martinucci, B., Paraggio, P. et al. Some Results on the Telegraph Process Confined by Two Non-Standard Boundaries. Methodol Comput Appl Probab 23, 837–858 (2021). https://doi.org/10.1007/s11009-020-09782-1

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  • DOI: https://doi.org/10.1007/s11009-020-09782-1

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