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Introduction to the Theory of Diffusion Processes / Edition 1

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Brand new. We distribute directly for the publisher. Focusing on one of the major branches of probability theory, this book treats the large class of processes with continuous ... sample paths that possess the "Markov property". The exposition is based on the theory of stochastic analysis. The diffusion processes discussed are interpreted as solutions of It's stochastic integral equations. The book is designed as a self-contained introduction, requiring no background in the theory of probability or even in measure theory. In particular, the theory of local continuous martingales is covered without the introduction of the idea of conditional expectation. Krylov covers such subjects as the Wiener process and its properties, the theory of stochastic integrals, stochastic differential equations and their relation to elliptic and parabolic partial differential equations, Kolmogorov's equations, and methods for proving the smoothness of probabilistic solutions of partial differential equations. With many exercises and thought-provoking problems, this book would be an excellent text for a graduate course in diffusion processes and related subjects. Read more Show Less

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Overview

Focusing on one of the major branches of probability theory, this book treats the large class of processes with continuous sample paths that possess the ''Markov property''. The exposition is based on the theory of stochastic analysis. The diffusion processes discussed are interpreted as solutions of Ito's stochastic integral equations. The book is designed as a self-contained introduction, requiring no background in the theory of probability or even in measure theory. In particular, the theory of local continuous martingales is covered without the introduction of the idea of conditional expectation. Krylov covers such subjects as the Wiener process and its properties, the theory of stochastic integrals, stochastic differential equations and their relation to elliptic and parabolic partial differential equations, Kolmogorov's equations, and methods for proving the smoothness of probabilistic solutions of partial differential equations. With many exercises and thought-provoking problems, this book would be an excellent text for a graduate course in diffusion processes and related subjects.

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Editorial Reviews

Booknews
Presents the basic elements of the theory of diffusion processes, for readers with no knowledge of probability theory or stochastic processes. Prerequisites are second-year university mathematics and a knowledge of one-dimensional Lebesgue measures. The diffusion processes are interpreted as solutions of Ito's stochastic integral equations. Translated from an original Russian manuscript. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

Table of Contents

Preface
Ch. I Elements of Measure and Integration Theory 1
1 Measurable spaces and random variables 1
2 Probability spaces, expectations 6
3 Completion of measure spaces. Relation between Riemann and Lebesgue integrals 14
4 Distributions of random elements, Gaussian vectors, independence 16
5 Lemma on [pi]- and [gamma]-systems; applications. Fubini's theorem 22
Ch. II The Wiener Process 27
1 Brownian motion and the Wiener process 27
2 Existence of the Wiener process 30
3 Some properties of the Wiener process 36
4 Multidimensional Wiener processes. Markov times 43
5 Strong Markov property of the Wiener process 46
6 Martingale properties of the Wiener process 50
7 Burkholder-Davis-Gundy inequalities and Wald identities for the Wiener process 54
8 The Wiener process and the heat equation. Martingales 57
9 Some applications of Theorem 8.6 64
10 The Wiener process and the Laplace operator 71
Ch. III Ito's Stochastic Integral 77
1 Integral with respect to a Random Orthogonal Measure 78
2 Ito's stochastic integral with respect to a Wiener process 84
3 Ito's stochastic integral with variable upper limit 90
4 Extending the set of Ito-integrable functions. The notion of a local martingale 95
5 Quadratic variation of stochastic integral and pseudopredictable functions 101
6 Passage to a limit within the Ito stochastic integral. Ito's inequalities. Convergence in probability 105
7 Ito's integral with respect to a multidimensional Wiener process 113
8 Ito's formula 115
9 Martingale version of Ito's formula. Levy's theorem 122
10 Stochastic integral with respect to an admissible local martingale 127
11 Regularly measurable processes 134
Ch. IV Some Applications of Ito's Formula 141
1 Transformation of Ito's formula; particular cases 141
2 Random time change in stochastic integrals 147
3 Girsanov's theorem 153
4 Burkholder-Davis-Gundy inequalities for multidimensional random processes 160
Ch. V Ito's Stochastic Equations 165
1 Existence and uniqueness of solutions of Ito stochastic equations 166
2 Two examples of application of Ito stochastic equations 173
3 Equations solvable by Euler's method 178
4 Some properties of Euler's approximations 182
5 Strong Markov property of solutions of stochastic equations 189
6 The Kolmogorov equations 195
7 Derivation of the Kolmogorov equation in the inhomogeneous case 200
8 Derivation of the Kolmogorov equations in the homogeneous case 206
9 Probabilistic solutions of partial differential equations 212
10 Proof of Theorem 9.4 215
Ch. VI Further Methods for Investigating the Smoothness of Probabilistic Solutions of Differential Equations 221
1 Some generalizations of Theorems V.8.1 and V.8.5 222
2 Quasiderivatives of solutions of stochastic equations 228
3 Proofs of Lemmas 1.3 and 1.8 232
4 Some ideas from the theory of conditional processes 236
5 A method for investigating the function (V.6.13) for [actual symbol not reproducible] 246
Appendix A. Proof of Lemma II.2.4 257
Appendix B. Proof of Theorem II.8.1 259
List of Notations 261
Comments 263
References 267
Index 269
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