# zbMATH — the first resource for mathematics

Anticipative direct transformations on the Poisson space. (English) Zbl 1021.60044
The author considers a random transformation of Poisson distributed clouds $$\omega$$ in $$U=[0,1]\times ({\mathbb R}^d\setminus \{0\})$$ by addition or deletion of points according to another Poisson cloud $$\tilde{\omega}$$ whose intensity depends itself on $$\omega$$ through a process $$f:U\times \Omega \to {\mathbb R}_+$$. Using a notion of time direction in $$U$$, this transformation gives rise to a stopped transformation $$Y_t$$ at each time $$t\geq 0$$. It is shown that $$(Y_t)_{t\in {\mathbb R}_+}$$ can be represented as a Markov process with values in the set of transformations that add or remove particles in Poisson clouds, according to a suitable rate process $$(h_u)_{u\in{\mathbb R}_+}$$. The absolute continuity of the final transformation $$Y_\infty$$ is proved when the transformed cloud is given by an adapted intensity process $$f_u(\omega)$$ and a possibly anticipating but finite component, which is dealt with using results of J. Picard [Ann. Inst. Henri Poincaré, Probab. Stat. 32, 509-548 (1996; Zbl 0859.60045)]. Several examples of applications are considered, including the perturbation of an $$\alpha$$-stable process by another stable process.
##### MSC:
 60H07 Stochastic calculus of variations and the Malliavin calculus 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60G51 Processes with independent increments; Lévy processes
Full Text: