Summary
Let
be a real-valued stochastic process having a continuous local timeL(u,t),u∈ —, 0≦t≦T andX ε(t) = (Ψ ε *X)(t),t ⪴ 0, the regularization ofX by means of the convolution with the approximation of unityΨ ε. 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 numberN X ε (u;•) of crossings of the processX ε 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.
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Nualart, D., Wschebor, M. Integration par parties dans l'espace de Wiener et approximation du temps local. Probab. Th. Rel. Fields 90, 83–109 (1991). https://doi.org/10.1007/BF01321135
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DOI: https://doi.org/10.1007/BF01321135