Joint Hitting-Time Densities for Finite State Markov Processes

TL;DR

This paper derives explicit formulas for the joint distribution of hitting times in finite state Markov processes, with applications to credit risk modeling and numerical examples.

Contribution

It introduces recursive formulas for joint densities and tail probabilities of hitting times, generalizing Poisson process jump time formulas.

Findings

Derived explicit joint density formulas for hitting times.

Provided recursive methods for tail probability calculations.

Illustrated relevance to credit risk modeling.

Abstract

For a finite state Markov process and a finite collection $\{ \Gamma_k, k \in K \}$ of subsets of its state space, let $\tau_k$ be the first time the process visits the set $\Gamma_k$. We derive explicit/recursive formulas for the joint density and tail probabilities of the stopping times $\{ \tau_k, k \in K\}$. The formulas are natural generalizations of those associated with the jump times of a simple Poisson process. We give a numerical example and indicate the relevance of our results to credit risk modeling.