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System Control and Rough Paths

Terry Lyons and Zhongmin Qian

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198506485
eISBN:
9780191709395
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506485.001.0001
Subject:
Mathematics, Probability / Statistics

Systems evolve and interact, and when they do this in a continuous way, differential equations can be used to provide accurate mathematical models for the interaction or reaction of the controlled ... More


ROUGH PATHS

Terry Lyons and Zhongmin Qian

in System Control and Rough Paths

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198506485
eISBN:
9780191709395
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506485.003.0003
Subject:
Mathematics, Probability / Statistics

This chapter defines rough paths. If the path has good smoothness properties, then its chords (the Abelian version of the description) provide an adequate description and can be used to predict its ... More


INTRODUCTION

Terry Lyons and Zhongmin Qian

in System Control and Rough Paths

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198506485
eISBN:
9780191709395
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506485.003.0001
Subject:
Mathematics, Probability / Statistics

The concept of a differential equation controlled by a rough path can be motivated by quite simple examples. One such example is a linear system driven by two-dimensional noise. This example is ... More


BROWNIAN ROUGH PATHS

Terry Lyons and Zhongmin Qian

in System Control and Rough Paths

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198506485
eISBN:
9780191709395
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506485.003.0004
Subject:
Mathematics, Probability / Statistics

This chapter proves that the dyadic polygonal approximations to a Brownian paths, with probability 1, a Cauchy sequence of rough paths, and in this way, the chapter proves a result first established ... More


PATH INTEGRATION ALONG ROUGH PATHS

Terry Lyons and Zhongmin Qian

in System Control and Rough Paths

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198506485
eISBN:
9780191709395
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506485.003.0005
Subject:
Mathematics, Probability / Statistics

This chapter establishes an integration theory for rough paths. The key result is the existence of the integral of a Lipschitz one-form against a rough path. In particular, this gives a change of ... More


VECTOR FIELDS AND FLOW EQUATIONS

Terry Lyons and Zhongmin Qian

in System Control and Rough Paths

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198506485
eISBN:
9780191709395
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506485.003.0007
Subject:
Mathematics, Probability / Statistics

This chapter studies spaces of paths as geometric objects in their own right, and particularly looks at the possibility of constructing interesting and mathematically meaningful vector fields on ... More


UNIVERSAL LIMIT THEOREM

Terry Lyons and Zhongmin Qian

in System Control and Rough Paths

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198506485
eISBN:
9780191709395
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506485.003.0006
Subject:
Mathematics, Probability / Statistics

The key result presented in this book and the result which justifies the definition of a rough path is the universal limit theorem. The chapter proves in section 6.3 that a system of differential ... More


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