Quasi sure quadratic variation of local times of smooth semimartingales.
Let \(L^x_t\) denote the local time of a continuous semimartingale \((M_t)\). Let \(\Delta_n= (a_{n,i}, a_{n,i+1})\) be any sequence of partitions of \([a,b]\) with mesh size converging to zero. It was shown by \textit{N. Bouleau} and \textit{M. Yor} [C. R. Acad. Sci., Paris, Sér. I 292, 491--494 (1981; Zbl 0476.60046)] that \[ \lim_{n\to\infty} \sum_{\Delta_n} (L^{a_{n,i+1}}_t- L^{a_{n,i}}_t)^2= 4\int^b_a L^x t \,dx\;\text{ in probability}. \tag{+} \] In the present paper it is shown for continuous semimartingales \((M_t)\) which are smooth in the sense of \textit{P. Malliavin} and \textit{D. Nualart} [J. Funct. Anal. 112, No. 2, 287--317 (1993; Zbl 0783.60006)] that the convergence in \((+)\) even holds almost surely for partitions \(\Delta_n= (a_{n,i}, a_{n,i+1})\) such that \(a_{n,i}= i(b- a)/2^n\), \(i= 0,1,\dots, 2^n\).
scientific article; zbMATH DE number 2055854
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