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On a time-changed variant of the generalized counting process

Part of: Stochastic processes

Published online by Cambridge University Press:  27 October 2023

M. Khandakar  and
K. K. Kataria
M. Khandakar*
Affiliation:
Indian Institute of Technology Bombay
K. K. Kataria*
Affiliation:
Indian Institute of Technology Bhilai
*
*Postal address: Department of Mathematics, Indian Institute of Technology Bombay, Mumbai 400076, India. Email address: mostafizar@math.iitb.ac.in
**Postal address: Department of Mathematics, Indian Institute of Technology Bhilai, Raipur 492015, India. Email address: kuldeepk@iitbhilai.ac.in
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Abstract

In this paper, we time-change the generalized counting process (GCP) by an independent inverse mixed stable subordinator to obtain a fractional version of the GCP. We call it the mixed fractional counting process (MFCP). The system of fractional differential equations that governs its state probabilities is obtained using the Z transform method. Its one-dimensional distribution, mean, variance, covariance, probability generating function, and factorial moments are obtained. It is shown that the MFCP exhibits the long-range dependence property whereas its increment process has the short-range dependence property. As an application we consider a risk process in which the claims are modelled using the MFCP. For this risk process, we obtain an asymptotic behaviour of its finite-time ruin probability when the claim sizes are subexponentially distributed and the initial capital is arbitrarily large. Later, we discuss some distributional properties of a compound version of the GCP.

Keywords

Mixed fractional Poisson processinverse mixed stable subordinatorlong-range dependence propertyrisk process

MSC classification

Primary: 60G22: Fractional processes, including fractional Brownian motion
Secondary: 60G55: Point processes
Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 2 , June 2024 , pp. 716 - 738
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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