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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.
This book is split into four parts. Part I is concerned with basic material about certain ordinary differential equations, paths of Hölder and variation regularity, and the rudiments of Riemann–Stieltjes and Young integration. Nothing here will be new to specialists, but the material seems rather spread out in the literature and we hope it will prove useful to have it collected in one place.
Part II is about the deterministic core of rough path theory, à la T. J. Lyons, but actually inspired by the direct approach of A. M. Davie. Although the theory can be formulated in a Banach setting, we have chosen to remain in a finitedimensional setting; our motivation for this decision comes from the fact that the bulk of classic texts on Brownian motion and stochastic analysis take place in a similar setting, and these are the grounds on which we sought applications.
In essence, with rough paths one attempts to take out probability from the theory of stochastic differential equations – to the extent possible. Probability still matters, but the problems are shifted from the analysis of the actual SDEs to the analysis of elementary stochastic integrals, known as Lévy's stochastic area. In Part III we start with a detailed discussion of how multidimensional Brownian motion can be turned into a (random) rough path; followed by a similar study for (continuous) semi-martingales and large classes of multidimensional Gaussian – and Markovian – processes.