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)
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.
scientific article; zbMATH DE number 20054
scientific article