A computational approach to extreme values and related hitting probabilities in level-dependent quasi-birth-death processes

This paper analyzes the dynamics of a level-dependent quasi-birth-death process 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 Mᵢ₀ and i₀, 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 X before its first visit to states in $l(0)$, τₘₐₓ is the first time that the level process (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 ₘₐₓ=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.