D∞-APPROXIMATION OF QUADRATIC VARIATIONS OF SMOOTH ITO PROCESSES

The purpose of this paper is to prove that the quadratic variations of smooth Ito process inthe sense of Malliavin-NuaJart can he approximated in Sobolev spaces over the Wiener space by its discrete quadratic variatlons.     

D∞-APPROXIMATION OF PRODUCT VARIATIONS OF TWO PARAMETER SMOOTH SEMIMARTINGALES

Let X be a two parameter smooth semimartingale and X~ be its process ofthe product variation. It is proved that X~ can be approximated as D∞-limit of sums ofits discrete product variations as the mesh of division tends to zero. Moreover, this resultcan be strengthen to yield the quasi sure convergence of sums by estimating the speed ofthe convergence.