Calculus for multiplicative functionals, Itô’s formula and differential equations

Abstract

The theory of stochasic integrals and stochastic differential equations was established by K Ito [3, 4] (also see [2]). In past four decade years, Ito’s stochastic analysis has established for itself the central role in modern probability theory. Ito’s theory of stochastic differential equations has been one of the most important tools. However, Ito’s construction of stochastic integrals over Brownian motion possesses an essentially random characterization, and is meaningless for a single Brownian path. The Ito map obtained by solving Ito’s stochastic differential equations is nowhere continuous on the Wiener space.