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Calculus for multiplicative functionals, Itô’s formula and differential equations

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Itô’s Stochastic Calculus and Probability Theory

Abstract

The theory of stochasic integrals and stochastic differential equations was established by K Itô [3, 4] (also see [2]). In past four decade years, Itô’s stochastic analysis has established for itself the central role in modern probability theory. Itô’s theory of stochastic differential equations has been one of the most important tools. However, Itô’s construction of stochastic integrals over Brownian motion possesses an essentially random characterization, and is meaningless for a single Brownian path. The Itô map obtained by solving Itô’s stochastic differential equations is nowhere continuous on the Wiener space.

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References

  1. Chen, K.T.: Integration of paths, geometric invariants and generalized Baker-Hausdorff formula, Ann. of Math., 163–178, (1957).

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Author information

Authors and Affiliations

  1. Department of Mathematics, Imperial College of Science, Technology & Medicine, 180 Queen’s Gate, London, SW7 2BZ, UK

    T. J. Lyons & Z. M. Qian

Authors
  1. T. J. Lyons
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  2. Z. M. Qian
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Ritsumeikan University, 525-77, Kusatsu, Shiga, Japan

    Nobuyuki Ikeda (Professor) (Professor)

  2. Department of Mathematics, Graduate School of Science, Kyoto University, 606-01, Kyoto, Japan

    Shinzo Watanabe (Professor) (Professor)

  3. Department of Mathematics, Faculty of Fundamental Engineering, Osaka University, Toyonaka, Osaka, 560, Japan

    Masatoshi Fukushima (Professor) (Professor)

  4. Graduate School of Mathematics, Kyushu University, Fukuoka, 812, Japan

    Hiroshi Kunita (Professor) (Professor)

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© 1996 Springer-Verlag Tokyo

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Lyons, T.J., Qian, Z.M. (1996). Calculus for multiplicative functionals, Itô’s formula and differential equations. In: Ikeda, N., Watanabe, S., Fukushima, M., Kunita, H. (eds) Itô’s Stochastic Calculus and Probability Theory. Springer, Tokyo. https://doi.org/10.1007/978-4-431-68532-6_15

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  • DOI: https://doi.org/10.1007/978-4-431-68532-6_15

  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-68534-0

  • Online ISBN: 978-4-431-68532-6

  • eBook Packages: Springer Book Archive

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