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Multidimensional Stochastic Processes as Rough Paths
Theory and Applications


Part of Cambridge Studies in Advanced Mathematics

  • Date Published: February 2010
  • availability: Available
  • format: Hardback
  • isbn: 9780521876070

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About the Authors
  • Rough path analysis provides a fresh perspective on Ito's important theory of stochastic differential equations. Key theorems of modern stochastic analysis (existence and limit theorems for stochastic flows, Freidlin-Wentzell theory, the Stroock-Varadhan support description) can be obtained with dramatic simplifications. Classical approximation results and their limitations (Wong-Zakai, McShane's counterexample) receive 'obvious' rough path explanations. Evidence is building that rough paths will play an important role in the future analysis of stochastic partial differential equations and the authors include some first results in this direction. They also emphasize interactions with other parts of mathematics, including Caratheodory geometry, Dirichlet forms and Malliavin calculus. Based on successful courses at the graduate level, this up-to-date introduction presents the theory of rough paths and its applications to stochastic analysis. Examples, explanations and exercises make the book accessible to graduate students and researchers from a variety of fields.

    • A modern introduction made accessible to researchers from related fields
    • Provides many exercises and solutions to test the reader's understanding
    • Emphasizes applications to stochastic analysis and interactions with other areas of mathematics
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    Product details

    • Date Published: February 2010
    • format: Hardback
    • isbn: 9780521876070
    • length: 670 pages
    • dimensions: 235 x 160 x 38 mm
    • weight: 1.6kg
    • contains: 6 b/w illus. 100 exercises
    • availability: Available
  • Table of Contents

    The story in a nutshell
    Part I. Basics:
    1. Continuous paths of bounded variation
    2. Riemann-Stieltjes integration
    3. Ordinary differential equations (ODEs)
    4. ODEs: smoothness
    5. Variation and Hölder spaces
    6. Young integration
    Part II. Abstract Theory of Rough Paths:
    7. Free nilpotent groups
    8. Variation and Hölder spaces on free groups
    9. Geometric rough path spaces
    10. Rough differential equations (RDEs)
    11. RDEs: smoothness
    12. RDEs with drift and other topics
    Part III. Stochastic Processes Lifted to Rough Paths:
    13. Brownian motion
    14. Continuous (semi)martingales
    15. Gaussian processes
    16. Markov processes
    Part IV. Applications to Stochastic Analysis:
    17. Stochastic differential equations and stochastic flows
    18. Stochastic Taylor expansions
    19. Support theorem and large deviations
    20. Malliavin calculus for RDEs
    Part V. Appendix: A. Sample path regularity and related topics
    B. Banach calculus
    C. Large deviations
    D. Gaussian analysis
    E. Analysis on local Dirichlet spaces
    Frequently used notation

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    Multidimensional Stochastic Processes as Rough Paths

    Peter K. Friz, Nicolas B. Victoir

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  • Authors

    Peter K. Friz, University of Cambridge
    Peter K. Friz is a Reader in the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge. He is also a Research Group Leader at the Johann Radon Institute at the Austrian Academy of Sciences, Linz.

    Nicolas B. Victoir
    Nicolas B. Victoir works in quantitative research at JPMorgan in Hong Kong.

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