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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.10895 |
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| _version_ | 1866914871545692160 |
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| author | Di Crescenzo, Antonio Gómez-Corral, Antonio Taipe, Diana |
| author_facet | Di Crescenzo, Antonio Gómez-Corral, Antonio Taipe, Diana |
| contents | This paper analyzes the dynamics of a level-dependent quasi-birth-death process ${\cal X}=\{(I(t),J(t)): t\geq 0\}$, i.e., a bi-variate Markov chain defined on the countable state space $\cup_{i=0}^{\infty} l(i)$ with $l(i)=\{(i,j) : j\in\{0,...,M_i\}\}$, for integers $M_i\in\mathbb{N}_0$ and $i\in\mathbb{N}_0$, 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 $(τ_{\max},I_{\max},J(τ_{\max}))$, where $I_{\max}$ is the running maximum level attained by process ${\cal X}$ before its first visit to states in $l(0)$, $τ_{\max}$ is the first time that the level process $\{I(t): t\geq 0\}$ reaches the running maximum $I_{\max}$, and $J(τ_{\max})$ is the phase at time $τ_{\max}$. Our methods rely on the use of restricted Laplace-Stieltjes transforms of $τ_{\max}$ on the set of sample paths $\{I_{\max}=i,J(τ_{\max})=j\}$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10895 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A computational approach to extreme values and related hitting probabilities in level-dependent quasi-birth-death processes Di Crescenzo, Antonio Gómez-Corral, Antonio Taipe, Diana Probability 60J28, 92B05 This paper analyzes the dynamics of a level-dependent quasi-birth-death process ${\cal X}=\{(I(t),J(t)): t\geq 0\}$, i.e., a bi-variate Markov chain defined on the countable state space $\cup_{i=0}^{\infty} l(i)$ with $l(i)=\{(i,j) : j\in\{0,...,M_i\}\}$, for integers $M_i\in\mathbb{N}_0$ and $i\in\mathbb{N}_0$, 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 $(τ_{\max},I_{\max},J(τ_{\max}))$, where $I_{\max}$ is the running maximum level attained by process ${\cal X}$ before its first visit to states in $l(0)$, $τ_{\max}$ is the first time that the level process $\{I(t): t\geq 0\}$ reaches the running maximum $I_{\max}$, and $J(τ_{\max})$ is the phase at time $τ_{\max}$. Our methods rely on the use of restricted Laplace-Stieltjes transforms of $τ_{\max}$ on the set of sample paths $\{I_{\max}=i,J(τ_{\max})=j\}$, 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. |
| title | A computational approach to extreme values and related hitting probabilities in level-dependent quasi-birth-death processes |
| topic | Probability 60J28, 92B05 |
| url | https://arxiv.org/abs/2407.10895 |