Q1177210
type: Item
instance of: scholarly article
title: Propriétés de martingales, explosion et représentation de Lévy- Khintchine d'une classe de processus de branchement à valeurs mesures. (Martingale properties, explosions and Levy-Khinchine representation of measure valued branching processes)
zbMATH Open document ID: 0743.60081
DOI: 10.1016/0304-4149(91)90093-R
author: Nicole El Karoui
author: Sylvie Rœlly
published in: Stochastic Processes and their Applications
publication date: 1992-06-26T00:00:00Z
review text: Let \(M(E)\) be a space of finite measures on a metric compact, endowed with the topology of weak convergence, \(B(E)\) be a space of measurable bounded functions on \(E\), \(B^ +(E)=\{f: f\in B(E),\quad \exists c>0, f\geq c\}\). It is considered a class of branching \(M(E)\)-valued Markov processes \((\Omega,{\mathcal F}_ t,X_ t,\zeta,P_ m)\) such that \[ Ep_ m[\mathbf{1}_{\{t<\zeta\}}e^{-\langle X_ t,f\rangle}]=e^{- \langle m,U_ tf\rangle}, t\geq 0,\quad m\in M(E), f\in B^ +(E), \] where the cumulant semigroup \(U_ t\), \(t\geq 0\), satisfies \(U_ tB^ +(E)\subseteq B^ +(E)\), \(t\geq 0\), and \[ U_ tf(x)=S_ tf(x)+\int^ t_ 0S_{t-s}R(\cdot,U_ s f(\cdot))(x)ds,\quad U_ 0f=f,\quad f\in B^ +(E), \] \(S_ t\) is a semigroup related to the right process on \(E\) and \[ R(x,z)=a(x)+b(x)z-{1\over 2}c(x)z^ 2-\int_{]0,\infty[}(e^{- \lambda z}-1+z\lambda\mathbf{1}_{\{0<\lambda\leq 1\}}\nu(x,d\lambda), x,z\in R_ +, \] satisfying the slight regularity assumptions. It is obtained the martingale characterization of \(X\) as a measure-valued semimartingale with the possible explosion and given local characteristics. A Girsanov type theorem is proved. It is also derived a Lévy-Khinchine representation formula for the Laplace transform of the probability law, corresponding to \(X\) on the Skorokhod space, and the related Lévy measure is characterized in terms of entrance law.
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60G44
zbMATH DE Number: 20054
zbMATH Keywords: measure-valued branching process
zbMATH Keywords: weak convergence
zbMATH Keywords: cumulant semigroup
zbMATH Keywords: martingale characterization
zbMATH Keywords: measure-valued semimartingale
zbMATH Keywords: local characteristics
zbMATH Keywords: Lévy-Khinchine representation formula
zbMATH Keywords: Laplace transform
zbMATH Keywords: Lévy measure
MaRDI profile type: Publication
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label: Propriétés de martingales, explosion et représentation de Lévy- Khintchine d'une classe de processus de branchement à valeurs mesures. (Martingale properties, explosions and Levy-Khinchine representation of measure valued branching processes)
description: scientific article; zbMATH DE number 20054