Multidimensional Stochastic Processes as Rough Paths: Theory and Applications

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.