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 appear here for the first time in book form, and the volume is sure to stimulate further research in this important field. The authors start with a detailed analysis of L?vy processes in infinite dimensions and their reproducing kernel Hilbert spaces; cylindrical L?vy 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.
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 appear here for the first time in book form, and the volume is sure to stimulate further research in this important field. The authors start with a detailed analysis of L?vy processes in infinite dimensions and their reproducing kernel Hilbert spaces; cylindrical L?vy 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.
Introduction; 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; Bibliography; Index.
Comprehensive monograph by two leading international experts; includes applications to statistical and fluid mechanics and to finance.
Szymon Peszat is an Associate Professor in the Institute of Mathematics at the 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).
"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
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