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Zbl 0744.60047
Azéma, J.; Hamza, K.
The property of predictable representation in the natural filtration of a regenerative set. (La propriété de représentation prévisible dans la filtration naturelle d'un ensemble régénératif.) (French)
[A] Séminaire de probabilités XXIII, Lect. Notes Math. 1372, 131-138 (1989).

[For the entire collection see Zbl 0722.00030.]\par Let $\Omega$ be a canonical space of closed subsets of $\bbfR\sb +$ of null Lebesgue measure. Let $H=\{(t,\omega)\in\bbfR\sb +\times\Omega:\ g\sp +\sb t(\omega)=t\}$, where $g\sp +\sb t(\omega)=\sup\{s\leq t:\ s\in\omega\}$, and let $({\cal F}\sb t)\sb{t\in\bbfR\sb +}$ be the natural filtration of $g\sp +\sb t$. Suppose that $P$ is a probability law on $\Omega$ such that $(\Omega,{\cal F}\sb t,H,P)$ is a perfect regenerative set. The authors show that the ${\cal F}\sb t$-martingale associated to the local time supported by $H$ has the predictable representation property. For related topics one may consult, e.g., the papers of {\it J. Jacod} and {\it J. Mémin} [Sémin. Probab. X, Univ. Strasbourg 1974/75, Lect. Notes Math. 511, 24-39 (1976; Zbl 0368.60058)] and of {\it J. Azéma} [Sémin. Probab. XIX, Univ. Strasbourg 1983/84, Proc., Lect. Notes Math. 1123, 397-495 (1985; Zbl 0563.60038)].

MSC 2000:: *60G44 Martingales with continuous parameter
60J55 Additive functionals

Keywords: perfect regenerative set; local time; predictable representation property

Citations: Zbl 0722.00030; Zbl 0368.60058; Zbl 0563.60038

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