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DIFFUSION LIMITS FOR A MARKOV MODULATED BINOMIAL COUNTING PROCESS

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  30 January 2019

Peter Spreij  and
Jaap Storm
Peter Spreij
Affiliation:
Korteweg-de Vries Institute for Mathematics, Universiteit van Amsterdam, Amsterdam, The Netherlands and IMAPP, Radboud University Nijmegen, Nijmegen, The Netherlands E-mail: spreij@uva.nl
Jaap Storm
Affiliation:
Department of Mathematics, Vrije Universiteit, Amsterdam, The Netherlands E-mail: p.j.storm@vu.nl
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Abstract

In this paper, we study limit behavior for a Markov-modulated binomial counting process, also called a binomial counting process under regime switching. Such a process naturally appears in the context of credit risk when multiple obligors are present. Markov-modulation takes place when the failure/default rate of each individual obligor depends on an underlying Markov chain. The limit behavior under consideration occurs when the number of obligors increases unboundedly, and/or by accelerating the modulating Markov process, called rapid switching. We establish diffusion approximations, obtained by application of (semi)martingale central limit theorems. Depending on the specific circumstances, different approximations are found.

Keywords

central limit theoremscounting processfunctional limit theoremsMarkov-modulated process

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60F05: Central limit and other weak theorems 60G99: None of the above, but in this section
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 34 , Issue 2 , April 2020 , pp. 235 - 257
Copyright
Copyright © Cambridge University Press 2019

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