A Comparison of Stochastic Integrals

Abstract

Two different stochastic integrals have been developed during the last ten years. One is largely associated with the work of E. J. McShane (the star integral), and the other has grown out of the work of H. Kunita and S. Watanabe (the dot integral). Assuming the customary conditions that guarantee the existence of the star integral, we give a formula relating the two integrals. We show that the star integral is equal to the dot integral provided one takes a projection of the integrand onto the space of predictable processes before evaluating the dot integral. This essentially embeds the theory of the star integral into that of the dot integral.