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Integration par parties dans l’espace de Wiener et approximation du temps local. (Integration by parts in the Wiener space and approximation of local time). (French) Zbl 0727.60052
Let \({\mathcal X}=(X_ t:\) \(t\in [0,T])\) be a real-valued stochastic process having a continuous local time L(u,t), \(u\in {\mathbb{R}}\), \(0\leq t\leq T\), and \(X_{\epsilon}(t)=(\Psi_{\epsilon}*X)(t)\), \(t\geq 0\), the regularization of X by means of the convolution with the approximation of unity \(\Psi_{\epsilon}\). The main theorem (Theorem 3.5) is a generalization of various results about the approximation (for fixed u) of the local time L(u,\(\cdot)\) by means of a convenient normalization of the number \(N^{X_{\epsilon}}(u;\cdot)\) of crossings of the process \(X_{\epsilon}\) with the level u. Especially, this theorem extends to a class of not necessarily Markovian continuous martingales, a result of this type for one-dimensional diffusions due to J. M. Azais [Ann. Inst. Henri Poincaré, Probab. Stat. 25, No.2, 175-194 (1989; Zbl 0674.60032)]. The methods of proof combine some estimations of the moments of the number of crossings with a level of a regular stochastic process with stochastic analysis techniques based upon integration by parts in the Wiener space.

60H05 Stochastic integrals
60J55 Local time and additive functionals
60G44 Martingales with continuous parameter
Full Text: DOI
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