Stochastic Differential Equations on Manifolds

K. D. Elworthy

• Publisher: Cambridge University Press
• Year: 1982
• Print ISBN:
  9780521287678
• eBook ISBN:
  9781107107830
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

The aims of this book, originally published in 1982, are to give an understanding of the basic ideas concerning stochastic differential equations on manifolds and their solution flows, to examine the properties of Brownian motion on Riemannian manifolds when it is constructed using the stochiastic development and to indicate some of the uses of the theory. The author has included two appendices which summarise the manifold theory and differential geometry needed to follow the development; coordinate-free notation is used throughout. Moreover, the stochiastic integrals used are those which can be obtained from limits of the Riemann sums, thereby avoiding much of the technicalities of the general theory of processes and allowing the reader to get a quick grasp of the fundamental ideas of stochastic integration as they are needed for a variety of applications.

Table of contents

• Cover
• Series Page
• Title
• Copyright
• CONTENTS
• PREFACE
• INTRODUCTION
• CHAPTER I : PRELIMINARIES AND NOTATION
• CHAPTER II : KOLMOGOROV'S THEOREM, TOTOKI'S THEOREM, AND BROWNIAN MOTION
• CHAPTER III : THE INTEGRAL: ESTIMATES AND EXISTENCE
• CHAPTER IV : SPECIAL CASES
• CHAPTER V : THE CHANGE OF VARIABLE FORMULA
• CHAPTER VI : STOCHASTIC INTEGRAL EQUATIONS
• CHAPTER VII : STOCHASTIC DIFFERENTIAL EQUATIONS ON MANIFOLDS
• CHAPTER VIII : REGULARITY
• CHAPTER IX : DIFFUSIONS
• APPENDIX A: MANIFOLDS AND FIBRE BUNDLES
• APPENDIX B: SOME DIFFERENTIAL GEOMETRY USING THE FRAME BUNDLE
• APPENDIX C SOME MEASURE THEORETIC TECHNICALITIES
• REFERENCES
• INDEX
• NOTATION AND ABBREVIATIONS