From Tanaka’s Formula to Ito’s Formula: Distributions, Tensor Products and Local Times

Abstract

In this article we study the classical finite dimensional Ito formula from an infinite dimensional perspective. A finite dimensional semi-martingale is represented as a semi-martingale in a (countable) Hilbert space of tempered distributions. The classical Ito formula is obtained on action by a test function from the dual space. Finite dimensional stochastic differential equations with smooth coefficients are represented as an SDE in a Hilbert space. We obtain representations of the local time process, viewed as a distribution in the space varible, in terms of a Hilbert space valued process of finite variation. A basic feature of our representation, is the role of the tensor product.