Stochastic Calculus in Manifolds
Addressed to both pure and applied probabilitists, including graduate students, this text is a pedagogically-oriented introduction to the Schwartz-Meyer second-order geometry and its use in stochastic calculus. P.A. Meyer has contributed an appendix: "A s
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Michel Emery
Stochastic Calculus in Manifolds With an Appendix by P. A. Meyer
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong
Michel Emery Paul-Andre Meyer Universite Louis Pasteur UER de Mathematiques et Informatique 7, rue Rene Descartes F-67084 Strasbourg Cedex France The cover picture depicts the martingale X" defmed in Exercise 4.11 on page 35.
Mathematics Subject Classification (1980): 60Hxx 58G32 60G48 53A99
ISBN-13: 978-3-540-51664-4 e-ISBN-13: 978-3-642-75051-9 DOL: 10.1007/978-3-642-75051-9 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1989 2141/3140-543210 Printed on acid-free paper
Ainsi, joignant la rigueur des demonstrations de la science a l'incertitude du hasard, et conciliant ces choses en apparence contraires, eUe peut, tirant son nom des deux, s'arroger a bon droit ce titre stupe£iant : La Geometrie du Hasard. B. Pascal, Adresse
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Contents
I. Real semimartingales and stochastic integrals Filtration, Process, Predictable, (1.1). Stopping time, Stochastic interval, Stopped process, (1.2). Convergence in probability uniformly on compact sets, Subdivision, Size of a subdivision, (1.3). Change of time, (1.4). Martingale, Continuous local martingale, Process with finite variation, Semimartingale, Local submartingale, Semimartingale up to infinity, (1.5). Locally bounded, Stochastic integral, (1.6). Quadratic variation of semimartingales, (1.7). Change of variable formula, (1.10). Stratonovich integral, (1.12). Existence, uniqueness and stability for the solution to a stochastic differential equation, (1.16,17).
II. Some vocabulary from differential geometry Manifold, (2.1). Whitney's imbedding theorem, (2.2). Tangent vector, Tangent space, (2.3). Push-forward of a vector, (2.6). Speed of a curve, (2.7). Tangent manifold, Vector field, (2.10). Cotangent vector, Covector, Form at a given point, (2.14). Form, (2.15). Pull-back of a form, (2.18). Bilinear form, (2.20). Pull-back of a bilinear form, (2.24). Flow of a vector field, Lie-derivative of a function, (2.25). Lie-derivative of a vector field, Commutator of two vector fields, (2.26). Lie-derivative of a form, (2.30). Lie-derivative of a bilinear form, (2.32).
III. Manifold-valued semimartingales and their quadratic variation AI-valued semimartingale, (3.1). Localness of AI-valued semimartingales, (3.4). Space-localness implies time-localness, (3.5). NI-valued semimartingale in an interval, AI-\'alued semimartingale up to infinity, (3.7)
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