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Analysis of Reflected Diffusions via an Exponential Time-Based Transformation

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Abstract

Let X(t) be a time-homogeneous diffusion process with state-space \([0,+\infty )\), where 0 is a reflecting or entrance endpoint, and let Z denote a random variable that describes the process X(t) evaluated at an exponentially distributed random time. We propose a method to obtain closed-form expressions for the conditional density and the mean of a new diffusion process Y(t), with the same state-space and with the same infinitesimal variance, whose drift depends on the infinitesimal moments of X(t) and on the hazard rate function of Z. This method also allows us to obtain the Laplace transform of the first-passage-time density of Y(t) through a lower constant boundary. We then discuss the relation between Y(t) and the process X(t) subject to catastrophes, as well as the interpretation of Y(t) as a diffusion in a decreasing potential. We study in detail some special cases concerning diffusion processes obtained when X(t) is the Wiener, Ornstein–Uhlenbeck, Bessel and Rayleigh process.

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Acknowledgments

The authors thank two anonymous referees for useful remarks and suggestions that improved the paper. Paper partially supported by GNCS-INdAM and Regione Campania (Legge 5).

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Authors and Affiliations

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

    Antonio Di Crescenzo

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

    Virginia Giorno & Amelia G. Nobile

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  1. Antonio Di Crescenzo
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Correspondence to Antonio Di Crescenzo.

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Di Crescenzo, A., Giorno, V. & Nobile, A.G. Analysis of Reflected Diffusions via an Exponential Time-Based Transformation. J Stat Phys 163, 1425–1453 (2016). https://doi.org/10.1007/s10955-016-1525-9

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