Explicit computations for some Markov modulated counting processes

TL;DR

This paper provides elementary, analytical computations for Markov modulated counting processes with regime switching, focusing on their conditional probabilities, characteristic functions, and limit behaviors under rapid switching, with an expository approach.

Contribution

It offers explicit formulas and limit results for counting processes with Markovian regime switching, enhancing understanding of their probabilistic structure and applications.

Findings

Conditional characteristic functions derived from probabilities

Limit behavior analyzed for rapid switching regimes

Processes applicable to default modeling and economic states

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.