TOWARD A STOCHASTIC CALCULUS, I^*

Abstract

Differential equations, deduced by physical theory for systems subjected to practically possible (Lipschitzian) random disturbances, can also be solved for martingale (e.g., Brownian-motion) disturbances by interpreting integrals as Itô integrals. But the two theories are not unified, and disconcerting differences appear. In this note and the next, we present the foundations of a unified theory, developed far enough to eliminate the discrepancies. Here we develop the tools; stochastic integration is extended to apply to a class of random functions including Lipschitzian and Brownian-motion processes, and a new kind of integral, called “doubly stochastic,” is defined and shown to exist under adequately general hypotheses.