Stochastic Partial Differential Equations with Lévy Noise
An Evolution Equation Approach

Comprehensive monograph by two leading international experts; includes applications to statistical and fluid mechanics and to finance.

S. Peszat (Author), J. Zabczyk (Author)

9780521879897, Cambridge University Press

Hardback, published 11 October 2007

432 pages
23.5 x 16.5 x 2.9 cm, 0.78 kg

'Summarising, this book is an excellent addition to the literature on stochastic partial differential equations in general and in particular with respect to evolution equations driven by a discontinuous noise. The exposition is self-contained and very well written and, in my opinion, will become a standard tool for everyone working on stochastic evolution equations and related areas.' Zentralblatt MATH

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.

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.

Subject Areas: Stochastics [PBWL], Calculus & mathematical analysis [PBK]