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Letters in Mathematical Physics

, Volume 28, Issue 2, pp 123–137 | Cite as

Stochastic differential calculus, the Moyal *-product, and noncommutative geometry

  • Aristophanes Dimakis
  • Folkert Müller-Hoissen
Article

Abstract

A reformulation of the Itô calculus of stochastic differentials is presented in terms of a differential calculus in the sense of noncommutative geometry (with an exterior derivative operator d satisfying d2 = 0 and the Leibniz rule). In this calculus, differentials do not commute with functions. The relation between both types of differential calculi is mediated by a generalized Moyal *-product. In contrast to the Itô calculus, the new framework naturally incorporates analogues of higher-order differential forms. A first step is made towards an understanding of their stochastic meaning.

Mathematics Subject Classifications (1991)

17B37 58A10 58G32 81R50 81S20 

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Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Aristophanes Dimakis
    • 1
  • Folkert Müller-Hoissen
    • 2
  1. 1.Department of MathematicsUniversity of CreteIraklionGreece
  2. 2.Institut für Theoretische PhysikGöttingenGermany

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