  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.
