# DIFFUSION LIMITS FOR A MARKOV MODULATED BINOMIAL COUNTING PROCESS

Peter Spreij, P.J. Storm

### 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.
Original language English 23 Probability in the Engineering and Informational Sciences https://doi.org/10.1017/S0269964818000578 Published - 30 Jan 2019

### Fingerprint

Diffusion Limit
Counting Process
Limit Behavior
Markov processes
Martingale Central Limit Theorem
Regime Switching
Credit Risk
Diffusion Approximation
Semimartingale
Markov Process
Markov chain
Modulation
Approximation
Counting process
Context
Markov process
Central limit theorem
Regime switching
Diffusion approximation
Credit risk

### Keywords

• Markov-modulated process
• central limit theorems
• counting process
• functional limit theorems

### Cite this

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title = "DIFFUSION LIMITS FOR A MARKOV MODULATED BINOMIAL COUNTING PROCESS",
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 = "Markov-modulated process, central limit theorems, counting process, functional limit theorems",
author = "Peter Spreij and P.J. Storm",
year = "2019",
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doi = "https://doi.org/10.1017/S0269964818000578",
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publisher = "Cambridge University Press",

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In: Probability in the Engineering and Informational Sciences, 30.01.2019.

TY - JOUR

T1 - DIFFUSION LIMITS FOR A MARKOV MODULATED BINOMIAL COUNTING PROCESS

AU - Spreij, Peter

AU - Storm, P.J.

PY - 2019/1/30

Y1 - 2019/1/30

N2 - 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.

AB - 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.

KW - Markov-modulated process

KW - central limit theorems

KW - counting process

KW - functional limit theorems

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U2 - https://doi.org/10.1017/S0269964818000578

DO - https://doi.org/10.1017/S0269964818000578

M3 - Article

JO - Probability in the Engineering and Informational Sciences

JF - Probability in the Engineering and Informational Sciences

SN - 0269-9648

ER -