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A two-sided stochastic integral and its calculus
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  • Published: September 1987

A two-sided stochastic integral and its calculus

  • E. Pardoux1 &
  • P. Protter2 

Probability Theory and Related Fields volume 76, pages 15–49 (1987)Cite this article

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  • 47 Citations

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Summary

Let X be a forward diffusion and Y a backward diffusion, both defined on [0,1], X t and Y tbeing respectively adapted to the past of a Wiener process W (·), and to its future increments. We construct a “two-sided” stochastic integral of the form.

$$\mathop \smallint \limits_0^t \Phi (u,X_u ,Y^u )dW(u)$$

which generalizes the backward and forward Itô integrals simultaneously. Our construction is quite intuitive, and leads to a generalized stochastic calculus. It is also shown that for each fixed t, our integral coincides with that defined by Skorohod in [18].

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Author information

Authors and Affiliations

  1. U.E.R. de Mathématiques, Université de Provence, 3,Place Victor Hugo, F-1331, Marseille Cedex 3, France

    E. Pardoux

  2. Department of Statistics, Purdue University, West Lafayette, IN, USA

    P. Protter

Authors
  1. E. Pardoux
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  2. P. Protter
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Additional information

Supported in part by NSF grant #DMS-8500997; part of the research for this work was performed while this author was visiting the Université de Provence at Marseille

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Pardoux, E., Protter, P. A two-sided stochastic integral and its calculus. Probab. Th. Rel. Fields 76, 15–49 (1987). https://doi.org/10.1007/BF00390274

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  • Received: 08 January 1986

  • Issue Date: September 1987

  • DOI: https://doi.org/10.1007/BF00390274

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Keywords

  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Wiener Process
  • Stochastic Calculus
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