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Stochastic Calculus

  • Leszek GawareckiEmail author
  • Vidyadhar Mandrekar
Chapter
  • 2.1k Downloads
Part of the Probability and Its Applications book series (PIA)

Abstract

After introducing cylindrical Gaussian random variables and Hilbert space valued Gaussian random variables, we define cylindrical Wiener process and Hilbert space valued Wiener process in a natural way. We develop the Itô stochastic integral with respect to cylindrical and Hilbert space valued Wiener processes simultaneously as they share many common features. Our construction of the Itô integral is consistent with the construction of a stochastic integral with respect to square integrable martingales, however, regarding measurability, we assume only that the integrand is adapted, following the classical ideas as presented by Liptser and Shiryayev (Statistics of Stochastic Processes. Nauka, 1974) and by Øksendal (Stochastic Differential Equations. Springer, New York, 1998). We provide detailed proofs of properties of the Itô integral and follow with the Martingale Representation Theorem. Our unified approach to stochastic integration with respect to cylindrical and Hilbert space valued Wiener processes allowed us to present the Stochastic Fubini Theorem and the Itô Formula in both cases in an almost identical way.

Keywords

Elementary Process Wiener Process Gaussian Random Variable Separable Hilbert Space Stochastic Calculus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 49.
    R. S. Liptzer and A. N. Shiryaev. Statistics of Stochastic Processes, Nauka, Moscow (1974). Google Scholar
  2. 61.
    B. Øksendal. Stochastic Differential Equations, Springer, New York (1998). Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of MathematicsKettering UniversityFlintUSA
  2. 2.Department of Statistics and ProbabilityMichigan State UniversityEast LansingUSA

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