Multidimensional Stochastic Processes as Rough Paths
Multidimensional Stochastic Processes as Rough Paths
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GBPRough 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
Product details
April 2010Adobe eBook Reader
9780511686511
0 pages
0kg
6 b/w illus. 100 exercises
This ISBN is for an eBook version which is distributed on our behalf by a third party.
Table of Contents
- Preface
- Introduction
- 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
- References
- Index.
