System control and rough paths / Terry Lyons and Zhongmin Qian.

Format:

Book

Author/Creator:

Lyons, Terry.

Contributor:

Qian, Zhongmin.

Alumni and Friends Memorial Book Fund.

Series:

Oxford mathematical monographs

Oxford science publications

Language:

English

Subjects (All):

Stochastic processes.

Physical Description:

x, 216 pages ; 24 cm.

Place of Publication:

Oxford : Clarendon Press ; New York : Oxford University Press, 2002.

Summary:

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 and the short to medium term evolution of a receiver. The core result of the book is a continuity theorum that proves that the response of the system depends continuously on these nilpotent elements.

Contents:

1.1.1 Controlled systems 5

1.1.2 Vector systems 5

1.1.3 Iterated integral expansions 7

1.2 Mathematics of rough paths 8

2 Lipschitz paths 11

2.2 Integration theory 16

2.3 Equations driven by Lipschitz paths 19

2.3.1 Existence of solutions 19

2.3.2 Uniqueness 21

2.3.3 Existence of solutions revisited 22

2.3.4 Continuity of the Ito map 23

3 Rough paths 28

3.1 Basic definitions and properties 28

3.1.1 The binomial inequality 31

3.1.2 Several basic results 35

3.2 Almost rough paths 40

3.3 Spaces of rough paths 47

3.3.1 Variation distances and variation topology 47

3.3.2 Young's integration theory 54

3.3.3 Elementary operations on rough paths 56

4 Brownian rough paths 61

4.1 Control variation distances 61

4.2 Dyadic polygonal approximations 67

4.3 Holder's condition 73

4.4 Processes with long-time memory 77

4.5 Gaussian processes 84

4.6 Wiener processes in Banach spaces 91

4.6.1 Gaussian analysis 91

4.6.2 Wiener processes as geometric rough paths 96

5 Path integration along rough paths 110

5.1 Lipschitz one-forms 110

5.2 Integration theory: degree two 117

5.3 Lipschitz continuity of integration 122

5.4 Ito's formula and stochastic integration 134

5.4.1 Ito's formula 134

5.4.2 Stochastic integration 135

5.5 Integration against geometric rough paths 137

5.6.1 Banach tensor products 144

5.6.2 Differentiation, Taylor's theorem 146

6 Universal limit theorem 148

6.2 Ito maps: rough paths with 2 [less than or equal] p [less than sign] 3 149

6.2.1 The Picard iteration 153

6.2.2 Basic estimates 155

6.2.3 Lipschitz continuity 158

6.2.4 Uniqueness 160

6.2.5 Continuity theorem 162

6.2.6 Flows of diffeomorphisms 165

6.3 The Ito map: geometric rough paths 168

7 Vector fields and flow equations 181

7.1 Smoothness of Ito maps 181

7.2 Ito's vector fields 186

7.3 Flows of Ito vector fields 191

7.4 Appendix: Driver's flow equation 201.

Notes:

Includes bibliographic references (pages 205-213) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

0198506481

OCLC:

51483643