2024/07/15 by Antonio Di Crescenzo, Di Crescenzo, Antonio, A. Gómez‐Corral +3 · 1 citation
Decision Sciences · Social Sciences · #60J28 #92B05 #FOS: Mathematics #Insurance, Mortality, Demography, Risk Management #Probability (math.PR) #Probability and Risk Models
paper · pdf · doi:10.48550/arxiv.2407.10895
openalex publication_date 2024/07/15 · openalex created_date 2024/07/17 · openalex updated_date 2026/07/28
This paper analyzes the dynamics of a level-dependent quasi-birth-death process \cal X=\(I(t),J(t)): t≥ 0\, i.e., a bi-variate Markov chain defined on the countable state space ∪i=0∞ l(i) with l(i)=\(i,j) : j∈\0,...,Mi\\, for integers Mi∈ℕ0 and i∈ℕ0, which has the special property that its q-matrix has a block-tridiagonal form. Under the assumption that the first passage to the subset l(0) occurs in a finite time with certainty, we characterize the probability law of (τmax,Imax,J(τmax)), where Imax is the running maximum level attained by process \cal X before its first visit to states in l(0), τmax is the first time that the level process \I(t): t≥ 0\ reaches the running maximum Imax, and J(τmax) is the phase at time τmax. Our methods rely on the use of restricted Laplace-Stieltjes transforms of τmax on the set of sample paths \Imax=i,J(τmax)=j\, and related processes under taboo of certain subsets of states. The utility of the resulting computational algorithms is demonstrated in two epidemic models: the SIS model for horizontally and vertically transmitted diseases; and the SIR model with constant population size.