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This book is available in Oxford Scholarship Online - view abstracts and keywords at book and chapter level.

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Applied Mathematics

Probability & Statistics

Signal Processing

System Control and Rough Paths

Terry Lyons and Zhongmin Qian

Clarendon Press Oxford Mathematical Monographs

228 pages | 234x156mm

978-0-19-850648-5 | Hardback | 19 December 2002

Price: £84.00

This item is printed to order. Items which are printed to order are normally despatched and charged within 5-10 days.

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A major research monograph
Provides a systematic account of pioneering work in probability and analysis
Highly original mathematical development in the solution of control problems
Allows for completely new classes of stochastic differential equations where the noise is no longer of Brownian or Poisson type

This book describes a completely novel mathematical development which has already influenced probability theory, and has potential for application to engineering and to areas of pure mathematics.

Intended for probabilists, mathematicians and engineers with a mathematical background from graduate level onwards, this book develops the evolution of complex non-linear systems subject to rough or rapidly fluctuating stimuli. Attention is focussed on an analysis of the relationship between the stimulus (or control) and the short to medium term evolution of a receiver (the response of the system).

A rapidly fluctuation stimuli can be likened to a huge dataset; and a basic question is how best to reduce this dataset so as to capture the critical information and little else. An essential component problem involves identifying the point at which two different stimuli produce essentially the same response from the class of receivers. (When do two stereo sounds sound the same?). This is an essentially non-linear problem that requires novel mathematics.

At one level, this book focuses on systems responding to such rough external stimuli, and demonstrates that the natural reduction approximates the stimuli as a sequence of nilpotent elements. The core result of the book is a continuity theorem that proves that the response of the system depends continuously on these nilpotent elements.

A key mathematical aspect of the book is the notion of a rough path, based on combining the notion of p-variation of Wiener with the iterated integral expansions of paths introduced by K. T. Chen. The continuity theorem for these rough paths gives a new way to construct solutions to stochastic differential equations, providing a fresh approach to the Itô theory but also allowing new kinds of noisy perturbations (such as Fractional Brownian Motions) that cannot be discussed in the standard Itô approach. It also provides some interesting concrete examples of 'continuous free groups'.

Readership: Graduate students and researchers in probability and mathematical control theory and engineers with a mathematical background.

The specification in this catalogue, including without limitation price, format, extent, number of illustrations, and month of publication, was as accurate as possible at the time the catalogue was compiled. Occasionally, due to the nature of some contractual restrictions, we are unable to ship a specific product to a particular territory. Jacket images are provisional and liable to change before publication.

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