Multidimensional Stochastic Processes as Rough Paths
Theory and Applications

An introduction to rough path theory and its applications to stochastic analysis, written for graduate students and researchers.

Peter K. Friz (Author), Nicolas B. Victoir (Author)

9780521876070, Cambridge University Press

Hardback, published 4 February 2010

670 pages, 6 b/w illus. 100 exercises
23.5 x 16 x 3.8 cm, 1.6 kg

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.

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.

Subject Areas: Calculus & mathematical analysis [PBK]