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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.

MSC:
 60H05 Stochastic integrals 60J55 Local time and additive functionals 60G44 Martingales with continuous parameter
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References:
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