Probabilistic analysis of systems alternating for state-dependent dichotomous noise

Antonio Di Crescenzo; Fabio Travaglino; Antonio Di Crescenzo; Fabio Travaglino

doi:10.3934/mbe.2019319

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 6386-6405. doi: 10.3934/mbe.2019319

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Probabilistic analysis of systems alternating for state-dependent dichotomous noise

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

Received: 30 December 2018 Accepted: 20 June 2019 Published: 10 July 2019

Aiming to construct a simple stochastic model able to describe systems alternating due to state-dependent dichotomous noise, we consider a generalized telegraph process whose sample-paths fluctuates around the zero state. Indeed, the latter process describes the motion of a particle on the real line, which is characterized by constant velocities and state-dependent intensities that vanish when the motion is toward the origin. This assumption allows to adopt an approach based on renewal theory to obtain formal expressions of the forward and backward transition densities of the process. The special case when certain random times of the motion possess gamma distribution leads to closed-form expressions of the transition densities, given in terms of the generalized Mittag-Leffler function. We also analyze a first-passage-time problem for the considered process in the presence of two constant boundaries.
- telegraph process,
- random motion,
- intensity function,
- interarrival times,
- gamma distribution,
- generalized Mittag-Leffler function,
- first-passage-time problem, constant boundaries
Citation: Antonio Di Crescenzo, Fabio Travaglino. Probabilistic analysis of systems alternating for state-dependent dichotomous noise[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 6386-6405. doi: 10.3934/mbe.2019319

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Abstract

Aiming to construct a simple stochastic model able to describe systems alternating due to state-dependent dichotomous noise, we consider a generalized telegraph process whose sample-paths fluctuates around the zero state. Indeed, the latter process describes the motion of a particle on the real line, which is characterized by constant velocities and state-dependent intensities that vanish when the motion is toward the origin. This assumption allows to adopt an approach based on renewal theory to obtain formal expressions of the forward and backward transition densities of the process. The special case when certain random times of the motion possess gamma distribution leads to closed-form expressions of the transition densities, given in terms of the generalized Mittag-Leffler function. We also analyze a first-passage-time problem for the considered process in the presence of two constant boundaries.

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