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McShane EJ. Toward a stochastic calculus, I. Proc Natl Acad Sci U S A. 1969;63(2):275-80doi: 10.1073/pnas.63.2.275.
McShane, E. J. (1969). Toward a stochastic calculus, I. Proceedings of the National Academy of Sciences of the United States of America, 63(2), 275-80. https://doi.org/10.1073/pnas.63.2.275
McShane, E J. "Toward a stochastic calculus, I." Proceedings of the National Academy of Sciences of the United States of America vol. 63,2 (1969): 275-80. doi: https://doi.org/10.1073/pnas.63.2.275
McShane EJ. Toward a stochastic calculus, I. Proc Natl Acad Sci U S A. 1969 Jun;63(2):275-80. doi: 10.1073/pnas.63.2.275. PMID: 16591757; PMCID: PMC223559.
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Proc Natl Acad Sci U S A. 1969 Jun;63(2):275-80. doi: 10.1073/pnas.63.2.275.

Toward a stochastic calculus, I.

Proceedings of the National Academy of Sciences of the United States of America

E J McShane

Affiliations

  1. UNIVERSITY OF VIRGINIA.

PMID: 16591757 PMCID: PMC223559 DOI: 10.1073/pnas.63.2.275

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.

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