Itô’s Formula for Non-Smooth Functions

Abstract

Itô’s formula, the chain rule in stochastic calculus, is going to be deduced in case of continuous functions possessing first and second order derivatives only in the sense of distributions. They are evaluated in Section 4.2 along the space-time Brownian motion and in Section 4.3 along non-degenerate locally Hölder-continuous space-time semimartingales which are local diffeomorphisms of a spatial parameter. This kind of results will be obtained by means of forward local C^1,ε-semimartingale flows of C¹-diffeomorphisms. First we derive a change of variable formula and show the existence of all moments of the involved Jacobian. Then our version of Itô’s formula can be established as a consequence of mollifier properties.