Markovian Trees Subject to Catastrophes: Transient Features and Extinction Probability

• Published in: Stochastic models Vol. 27; no. 4; pp. 569 - 590
• Main Authors: Hautphenne, Sophie, Latouche, Guy
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Taylor & Francis Group 01.10.2011; Taylor & Francis; Taylor & Francis Ltd
• Subjects: Branching processes; Catastrophes; Combinatorics; Combinatorics. Ordered structures; Exact sciences and technology; Extinction probability; Graph theory; Inference from stochastic processes; time series analysis; Markov analysis; Mathematics; Matrix analytic methods; Primary 60J22; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Secondary 60J80, 65C40; Statistics; Stochastic analysis; Stochastic processes; Studies; Transient measures; Markov process; Secondary 60J80; Stochastic model; Catastrophes; Primary 60J22; Probability distribution; Extinction probability; Multivariate analysis; Stochastic method; Stochastic process; Transients; Markov chain; Hypothesis test; Statistical test; Correspondence analysis; Arrival process; Integral equation; Matrix analytic methods; Factor analysis; Generating function; Extinction; Probability theory; Probability; Statistical estimation; Transient measures; Branching process; 65C40; Numerical analysis; Binary tree; Analytical method; Matrix method; Population size; Branching processes; Principal component analysis
• ISSN: 1532-6349; 1532-4214
• DOI: 10.1080/15326349.2011.614181

We study transient features and the extinction probability of a particular class of multi-type Markovian branching processes called Markovian binary trees, subject at random epochs to catastrophes, controlled by a Markovian arrival process, and killing random numbers of living individuals. We provide two types of probabilistic methods to numerically compute the extinction probability: the first one is based on an integral equation for the probability generating function of the population size, and the second one is based on the correspondence between Markovian branching processes with catastrophes and structured Markov chains of G/M/1-type.