An Intrinsic Proof of an Extension of Itô’s Isometry for Anticipating Stochastic Integrals

Abstract

Itô’s isometry forms the cornerstone of the definition of Itô’s integral and consequently the theory of stochastic calculus. Therefore, for any theory which extends Itô’s theory, it is important to know if the isometry holds. In this paper, we use probabilistic arguments to demonstrate that the extension of the isometry formula contains an extra term for the anticipating stochastic integral defined by Ayed and Kuo. We give examples to illustrate the usage of this formula and to show that the extra term can be positive or negative.