The Kurzweil-Henstock theory of stochastic integration

Abstract

The Kurzweil-Henstock approach has been successful in giving an alternative definition to the classical Ito integral, and a simpler and more direct proof of the Ito Formula. The main advantage of this approach lies in its explicitness in defining the integral, thereby reducing the technicalities of the classical stochastic calculus. In this note, we give a unified theory of stochastic integration using the Kurzweil-Henstock approach, using the more general martingale as the integrator. We derive Henstock’s Lemmas, absolute continuity property of the primitive process, integrability of stochastic processes and convergence theorems for the Kurzweil-Henstock stochastic integrals. These properties are well-known in the classical (non-stochastic) integration theory but have not been explicitly derived in the classical stochastic integration.