Quasi sure quadratic variation of smooth martingales
We prove that the quadratic variation process of a smooth martingale \({\mathbf M}= \{M_ t\}\) in the sense of Malliavin-Nualart can be constructed as the quasi-sure limit of sums of the form \(\sum M(\Delta_ i)\), as the size of the subdivision \(\Delta\) goes to zero. Then this result is extended to the case of smooth semimartingales and two- parameter strong smooth martingales. A recent work of \textit{Z. Liang} (preprint) shows that the same result holds true for two-parameter (not necessarily strong) smooth martingales.
quadratic variation process of a smooth martingale
scientific article; zbMATH DE number 633569