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

Format:

Book

Author/Creator:

Lyons, Terry.

Contributor:

Qian, Zhongmin.

Series:

Oxford mathematical monographs.

Oxford science publications.

Oxford mathematical monographs

Oxford science publications

Language:

English

Subjects (All):

Stochastic processes.

Mathematical statistics.

Physical Description:

1 online resource (227 p.)

Place of Publication:

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

Language Note:

English

Summary:

This work 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: the evolution of complex non-linear systems subject to rough or rapidly fluctuating stimuli.

Contents:

Contents; 1 Introduction; 1.1 Background and general description; 1.1.1 Controlled systems; 1.1.2 Vector systems; 1.1.3 Iterated integral expansions; 1.2 Mathematics of rough paths; 2 Lipschitz paths; 2.1 Several examples; 2.2 Integration theory; 2.3 Equations driven by Lipschitz paths; 2.3.1 Existence of solutions; 2.3.2 Uniqueness; 2.3.3 Existence of solutions revisited; 2.3.4 Continuity of the ltô map; 2.4 Comments and notes on Chapter 2; 3 Rough paths; 3.1 Basic definitions and properties; 3.1.1 The binomial inequality; 3.1.2 Several basic results; 3.2 Almost rough paths

3.3 Spaces of rough paths3.3.1 Variation distances and variation topology; 3.3.2 Young's integration theory; 3.3.3 Elementary operations on rough paths; 3.4 Comments and notes on Chapter 3; 4 Brownian rough paths; 4.1 Control variation distances; 4.2 Dyadic polygonal approximations; 4.3 Hölder's condition; 4.4 Processes with long-time memory; 4.5 Gaussian processes; 4.6 Wiener processes in Banach spaces; 4.6.1 Gaussian, analysis; 4.6.2 Wiener processes as geometric rough paths; 4.7 Comments and notes on Chapter 4; 5 Path integration along rough paths; 5.1 Lipschitz one-forms

5.2 Integration theory: degree two5.3 Lipschitz continuity of integration; 5.4 Itô's formula and stochastic integration; 5.4.1 Itô's formula; 5.4.2 Stochastic integration; 5.5 Integration against geometric rough, paths; 5.6 Appendix of Chapter 5; 5.6.1 Banach tensor products; 5.6.2 Differentiation, Taylor's theorem; 5.7 Comments and notes on Chapter 5; 6 Universal limit theorem; 6.1 Introduction; 6.2 Itô maps: rough paths with 2 < p < 3; 6.2.1 The Picard iteration; 6.2.2 Basic estimates; 6.2.3 Lipschitz continuity; 6.2.4 Uniqueness; 6.2.5 Continuity theorem; 6.2.6 Flows of diffeomorphisms

6.3 The Itô map: geometric rough paths6.4 Comments and notes on Chapter 6; 7 Vector fields and flow equations; 7.1 Smoothness of Itô maps; 7.2 Itô's vector fields; 7.3 Flows of Itô vector fields; 7.4 Appendix: Driver's flow equation; 7.5 Comments and notes on Chapter 7; Bibliography; Index; A; B; C; D; E; F; G; H; I; L; M; N; O; P; Q; R; S; T; U; W; Y

Notes:

Description based upon print version of record.

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

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

1-280-82942-7

9786610829422

0-19-152312-7

OCLC:

302357701