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A point process model with stochastic intensities for a branching population of two dependent types

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Eike Born
Eike Born*
Affiliation:
München
*
∗ Postal address: Ottostrasse 14, 85622 Feldkirchen, Germany. Email address: Eike.Born@mch.sni.de
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Abstract

We study a point process model with stochastic intensities for a particular branching population of individuals of two types. Type-I individuals immigrate into the population at the times of a Poisson process. During their lives they generate type-II individuals according to a random age dependent birth rate, which themselves may multiply and die. Living type-II descendants increase the death intensity of their type-I ancestor, and conversely, the multiplication and dying intensities of type-II individuals may depend on the life situation of their type-I ancestor. We show that the probability generating function of the marginal distribution of a type-I individual's life process, conditioned on its individual infection and death risk, satisfies an initial value problem of a partial differential equation, and derive its solution. This allows for the determination of additional distributions of observable random variables as well as for describing the complete population process.

Keywords

Two-type branching processdependent typespoint process modelrandom environmentstochastic intensitiespartial differential equations

MSC classification

Primary: 60J85: Applications of branching processes
Secondary: 60G55: Point processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 3 , September 1998 , pp. 723 - 739
Copyright
Copyright © Applied Probability Trust 1998 

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