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

In: Itô’s Stochastic Calculus and Probability Theory

Author

Listed:
  • T. J. Lyons

    (Imperial College of Science, Technology & Medicine, Department of Mathematics)

  • Z. M. Qian

    (Imperial College of Science, Technology & Medicine, Department of Mathematics)

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.

Suggested Citation

  • T. J. Lyons & Z. M. Qian, 1996. "Calculus for multiplicative functionals, Itô’s formula and differential equations," Springer Books, in: Nobuyuki Ikeda & Shinzo Watanabe & Masatoshi Fukushima & Hiroshi Kunita (ed.), Itô’s Stochastic Calculus and Probability Theory, pages 233-250, Springer.
    • Handle: RePEc:spr:sprchp:978-4-431-68532-6_15
    DOI: 10.1007/978-4-431-68532-6_15
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