Diffusion limits for a Markov modulated binomial counting process

TL;DR

This paper investigates the asymptotic behavior of a Markov-modulated binomial counting process, deriving diffusion approximations in scenarios with many obligors or rapid regime switching, relevant for credit risk modeling.

Contribution

It provides new diffusion limit results for Markov-modulated binomial processes under various scaling regimes, extending previous models in credit risk analysis.

Findings

Diffusion approximations derived using martingale central limit theorems.

Different limiting behaviors identified depending on scaling and switching speed.

Applicable to large portfolios with regime-dependent default rates.

Abstract

In this paper we study limit behavior for a Markov-modulated (MM) 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.