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

  • T. J. Lyons
  • Z. M. Qian


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.


Brownian Motion Stochastic Differential Equation Separable Banach Space Stochastic Integral Wiener Space 
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  1. 1.
    Chen, K.T.: Integration of paths, geometric invariants and generalized Baker-Hausdorff formula, Ann. of Math., 163–178, (1957).Google Scholar
  2. 2.
    Ikeda, N. & Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, North-Holland Pub. Corn., Amsterdam, Oxford, New York, Tokyo, (1981).MATHGoogle Scholar
  3. 3.
    Ito, K.: Stochastic integral, Proc. Imp. Acad. Tokyo, 20, 519–524, (1944).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Ito, K.: On stochastic differential equations, Mem. Amer. Math. Soc., 4 (1951).Google Scholar
  5. 5.
    Lyons, T.: Differential equations driven by rough signals (I): An extension of an inequality of L.C.Young, Math. Research Letters 1, 451–464 (1994).MathSciNetMATHGoogle Scholar
  6. 6.
    Lyons,T.: The interpretation and solution of ordinary differential equation driven by rough noise, Proc. of Symposia in Pure Math. vol. 57, (1995).Google Scholar
  7. 7.
    Lyons,T.: Differential equations driven by rough signals, (1995).Google Scholar
  8. 8.
    Young, L.C.: An inequality of Hölder type, connected with Stieltjes integration, Acta Math. 67, 251–282, (1936).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Tokyo 1996

Authors and Affiliations

  • T. J. Lyons
    • 1
  • Z. M. Qian
    • 1
  1. 1.Department of MathematicsImperial College of Science, Technology & MedicineLondonUK

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