Extension of Ito's Calculus via Malliavin Calculus

摘要

This work is devoted to the development of the Ito's calculus for a class of functionals defined on the Wiener space which are more general objects than semimartingales. In doing this, we begin by an extension if the Ito formula for finite dimensional hypoelliptic Ito processes to the tempered distributions. Watanabe has defined the composition of a tempered distribution by a hypoelliptic Wiener functional with the help of the Malliavin Calculus. Here we go a little further and give an Ito formula by using the same method. Let us note that when the Ito process is the standard Wiener process, the Ito formula has already been extended to the tempered distributions with the use of the Hida calculus; we give here a different approach which works for more general processes than the standard Wiener process. In the extended Ito formula, the Lebesgue integral part can be interpreted as a Bochner integral in some Sobolev space on the Weiner space and the notations; however the remaining part is not an ordinary stochastic integral, despite the fact that it corresponds to a functional in some Sobolev space on the Wiener space. This situation suggests an extension of the Ito stochastic integral to the objects which are not necessarily stochastic processes.