Explicit Computations for Some Markov Modulated Counting Processes

Abstract

In this paper we present elementary computations for some Markov modulated counting processes, also called counting processes with regime switching. Regime switching has become an increasingly popular concept in many branches of science. In finance, for instance, one could identify the background process with the 'state of the economy', to which asset prices react, or as an identification of the varying default rate of an obligor. The key feature of the counting processes in this paper is that their intensity processes are functions of a finite state Markov chain. This kind of processes can be used to model default events of some companies.Many quantities of interest in this paper, like conditional characteristic functions, can all be derived from conditional probabilities, which can, in principle, be analytically computed. We will also study limit results for models with rapid switching, which occur when inflating the intensity matrix of the Markov chain by a factor tending to infinity. The paper is largely expository in nature, with a didactic flavor.

Diffusion Limits for a Markov Modulated Binomial Counting Process

“…Proof: The first assertion is shown in, for instance, the recent reference Mandjes and Spreij [43], Corollary 2. For the second step we find ourselves in the situation of Proposition 4.1, and if we apply this result the proof is complete.…”

“…We call N the Markov-modulated binomial point process. See also Mandjes and Spreij [43] for further details on the construction of this process, and for a justification of the following reasonable assumption. There are also situations known where this assumption is violated by construction, see Spreij [50] for an example.…”

“…We call N the MM binomial point process. See also Mandjes and Spreij [43] for further details on the construction of this process, and for a justification of the following reasonable assumption.…”

“…Following the considerable interest in MM financial models we consider scaling limits, also referred to as diffusion approximations, of a MM model that has a natural interpretation in Credit Risk, (see Mandjes and Spreij [43]). In the basic setting there are n obligors which have independently exponential distributed default times τ i with intensity parameter λ > 0.…”