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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
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