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Non-homogeneous and time-changed versions of generalized counting processes

Part of: Stochastic processes

Published online by Cambridge University Press:  27 May 2025

Kuldeep Kumar Kataria ,
Mostafizar Khandakar  and
Palaniappan Vellaisamy
Kuldeep Kumar Kataria*
Affiliation:
Indian Institute of Technology Bhilai
Mostafizar Khandakar*
Affiliation:
Indian Institute of Information Technology Guwahati
Palaniappan Vellaisamy*
Affiliation:
University of California, Santa Barbara
*
*Postal address: Department of Mathematics, Indian Institute of Technology Bhilai, Durg-491002, India. Email: kuldeepk@iitbhilai.ac.in
**Postal address: Department of Science and Mathematics, Indian Institute of Information Technology Guwahati, Bongora-781015, India. Email: mostafizar.khandakar@iiitg.ac.in
***Postal address: Department of Statistics and Applied Probability, University of California, Santa Barbara, CA-93106, USA. Email: pvellais@ucsb.edu
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Abstract

In this paper, we introduce a non-homogeneous version of the generalized counting process (GCP). We time-change this process by an independent inverse stable subordinator and derive the system of governing differential–integral equations for the marginal distributions of its increments. We then consider the GCP time-changed by a multistable subordinator and obtain its Lévy measure and the distribution of its first passage times. We discuss an application of a time-changed GCP, namely the time-changed generalized counting process-I (TCGCP-I) in ruin theory. A fractional version of the TCGCP-I is studied, and its long-range dependence property is established.

Keywords

Bernštein functionmultistable subordinatornon-homogenous Poisson processlong-range dependence propertyrisk process

MSC classification

Primary: 60G22: Fractional processes, including fractional Brownian motion 60G55: Point processes
Secondary: 60G51: Processes with independent increments; L'evy processes

Information

Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 37
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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