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Stochastic Partial Differential Equations with Lévy Noise

Stochastic Partial Differential Equations with Lévy Noise
An Evolution Equation Approach

$154.00 (C)

Part of Encyclopedia of Mathematics and its Applications

  • Date Published: November 2007
  • availability: Available
  • format: Hardback
  • isbn: 9780521879897


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About the Authors
  • Recent years have seen an explosion of interest in stochastic partial differential equations where the driving noise is discontinuous. In this comprehensive monograph, two leading experts detail the evolution equation approach to their solution. Most of the results appeared here for the first time in book form. The authors start with a detailed analysis of Lévy processes in infinite dimensions and their reproducing kernel Hilbert spaces; cylindrical Lévy processes are constructed in terms of Poisson random measures; stochastic integrals are introduced. Stochastic parabolic and hyperbolic equations on domains of arbitrary dimensions are studied, and applications to statistical and fluid mechanics and to finance are also investigated. Ideal for researchers and graduate students in stochastic processes and partial differential equations, this self-contained text will also interest those working on stochastic modeling in finance, statistical physics and environmental science.

    • Was the first book to detail the evolution equation approach to the solution of stochastic partial differential equations with Lévy noise
    • Rapidly growing topic - majority of results appear here for the first time
    • Great potential for applications to finance, statistical mechanics and fluid dynamics
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    Reviews & endorsements

    "Peszat and Zabczyk (both are in the department of mathematics of the Polish Academy of Sciences) offer an important contribution to the literature on stochastic processes that will be of interest to graduate students and researchers. Their theory builds on the results of equations driven by Wiener processes and results of both L'evy and Wiener noise are discussed in tandem. Eight initial chapters provide a foundation to the theory that follows, with discussion that includes the basis of equations with L'evy noise, probability theory with martingales, L'evy processes and semigroups, cylindrical processes and reproducing kernels, and stochastic integration. Existence and regularity are explored in chapters that examine wave and delay equations, equations driven by spatially homogeneous noise, and equations with noise on the boundary, among other topics. The theory is then applied, in five chapters on invariant measures, Lattice systems, stochastic Burgers equation, an environmental pollution model, and in six bond market models. Several appendices provide a number of related proofs and results. A list of symbols is provided."
    Book News

    "... this volume represents a very important addition to the literature on stochastic partial differential equations driven by a discontinuous noise and provides an exhaustive unified treatment on topics which have attracted considerable attention in the last years. Moreover, the book makes a worthwhile effort to be accessible and self-contained and to compare step-by-step classical results obtained for SPDE's driven by the Wiener process with more recent results obtained in the case of the Levy process. For these reasons we are sure that the present book by Peszat and Zabczyk will be a valuable and unavoidable source for mathematicians who want to learn more about SPDE's with Levy noise.
    Sandra Cerrai, Mathematical Reviews

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    Product details

    • Date Published: November 2007
    • format: Hardback
    • isbn: 9780521879897
    • length: 432 pages
    • dimensions: 235 x 165 x 29 mm
    • weight: 0.78kg
    • availability: Available
  • Table of Contents

    Part I. Foundations:
    1. Why equations with Lévy noise?
    2. Analytic preliminaries
    3. Probabilistic preliminaries
    4. Lévy processes
    5. Lévy semigroups
    6. Poisson random measures
    7. Cylindrical processes and reproducing kernels
    8. Stochastic integration
    Part II. Existence and Regularity:
    9. General existence and uniqueness results
    10. Equations with non-Lipschitz coefficients
    11. Factorization and regularity
    12. Stochastic parabolic problems
    13. Wave and delay equations
    14. Equations driven by a spatially homogeneous noise
    15. Equations with noise on the boundary
    Part III. Applications:
    16. Invariant measures
    17. Lattice systems
    18. Stochastic Burgers equation
    19. Environmental pollution model
    20. Bond market models
    Appendix 1. Operators on Hilbert spaces
    Appendix 2. C0-semigroups
    Appendix 3. Regularization of Markov processes
    Appendix 4. Itô formulae
    Appendix 5. Lévy-Khinchin on [0,+ )
    Appendix 6. Proof of Lemma
    List of symbols

  • Authors

    S. Peszat, Polish Academy of Sciences
    Szymon Peszat is an Associate Professor in the Institute of Mathematics at the Polish Academy of Sciences.

    J. Zabczyk, Polish Academy of Sciences
    Jerzy Zabczyk is a Professor in the Institute of Mathematics at the Polish Academy of Sciences. He is the author (with G. Da Prato) of three earlier books for Cambridge University Press: Stochastic Equations in Infinite Dimensions (1992), Ergodicity for Infinite Dimensional Systems (1996) and Second Order Partial Differential Equations (2002).

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