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Multidimensional stochastic processes as rough paths. Theory and applications. (English) Zbl 1193.60053
Cambridge Studies in Advanced Mathematics 120. Cambridge: Cambridge University Press (ISBN 978-0-521-87607-0/hbk). xiv, 656 p. £ 50.00; $ 85.00 (2010).
This is a very detailed introduction to the theory of rough paths as developed by T. J. Lyons. It is one of the first monographs about this relatively new subject. The book will be appealing both to graduate students to get acquainted with the theory and to researchers as a reference.
The rough path approach gives a new perspective on the theory of stochastic integration that complements Ito’s classical theory: It enhances the paths of continuous functions. For example, instead of considering Brownian Motion as an \(\mathbb{R}^d\)-valued process, consider it as a path with values in the so-called free nilpotent group. This enlarged path carries enough information to allow for a deterministic (i.e., pathwise) treatment of stochastic differential equations.
The monograph is essentially self-contained. In the first part the authors introduce the deterministic prerequisites for the rough path theory: It treats continuous paths of bounded variation, Hölder continuity and \(p\)-variation as well as ordinary differential equations, Riemann-Stieltjes integration and Young integration. These classic subjects are of course well known, but as the authors point out “the material seems rather spread out in the literature”, so it is very convenient to have it included here.
The second part treats the deterministic theory of rough paths. The free nilpotent group is introduced and the connections to Lie group theory are pointed out. After studying Hölder-continuous paths with values in the free nilpotent group, the authors are ready to introduce and study geometric rough paths. Finally these results are applied to ordinary differential equations and eventually rough differential equations: The rough path theory allows to find unique solutions to differential equations of the type \[ dy = V(y) dx \] where \(x\) is a geometric rough path, and it automatically equips us with a flow.
The following part gives canonic lifts to geometric rough paths of a large number of stochastic processes: Brownian Motion, continuous semi-martingales, general Gaussian processes, and continuous Markov processes. This part will certainly be a very valuable reference for researchers.
The fourth and last section applies the previously derived results to stochastic analysis and serves as a demonstration of the potential of rough path theory: Stochastic flows are obtained without any further effort. The theory allows to treat stochastic differential equations driven by non-semi-martingales, or to treat anticipating integrands. Stochastic Taylor expansions, support theorems and large deviations are obtained as easy corollaries. However of course this comes with a price: All these results are proved with much less effort in the setting of rough path theory, but also the rough path approach gives less general results than the classical proofs, since here we need to impose stronger smoothness assumptions on the integrands.
Finally there is a large appendix, mostly to help the reader to get a quick overview on topics like large deviations, so that they can fully appreciate the applications of the rough path theory that are given in the previous chapters.

60G17 Sample path properties
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H05 Stochastic integrals
46N30 Applications of functional analysis in probability theory and statistics
60H07 Stochastic calculus of variations and the Malliavin calculus
60F10 Large deviations
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