Integration par parties dans l'espace de Wiener et approximation du temps local

Abstract

Let be a real-valued stochastic process having a continuous local timeL(u,t),u∈ —, 0≦t≦T andXe(t) = (Ψe *X)(t),t ⪴ 0, the regularization ofX by means of the convolution with the approximation of unityΨe. The main theorem in this paper (Theorem 3.5) is a generalization of various results about the approximation (for fixedu) of the local timeL(u, •) by means of a convenient normalization of the numberNXe(u;•) of crossings of the processXe with the levelu. Especially, this Theorem extends to a class of not necessarily Markovian continuous martingales, a result of this type for one-dimensional diffusions due to Azais [A2]). The methods of proof combine some estimations of the moments of the number of crossings with a level of a regular stochastic processes with stochastic analysis techniques based upon integration by parts in the Wiener space.