Introduction to the Theory of Diffusion Processes / Edition 1

Introduction to the Theory of Diffusion Processes / Edition 1

by N. V. Krylov
     
 

ISBN-10: 0821846000

ISBN-13: 9780821846001

Pub. Date: 03/01/1995

Publisher: American Mathematical Society

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

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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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Product Details

ISBN-13:
9780821846001
Publisher:
American Mathematical Society
Publication date:
03/01/1995
Series:
Translations of Mathematical Monographs Series, #14
Edition description:
New Edition
Pages:
271
Product dimensions:
7.48(w) x 10.63(h) x (d)

Table of Contents

Preface
Ch. IElements of Measure and Integration Theory1
1Measurable spaces and random variables1
2Probability spaces, expectations6
3Completion of measure spaces. Relation between Riemann and Lebesgue integrals14
4Distributions of random elements, Gaussian vectors, independence16
5Lemma on [pi]- and [gamma]-systems; applications. Fubini's theorem22
Ch. IIThe Wiener Process27
1Brownian motion and the Wiener process27
2Existence of the Wiener process30
3Some properties of the Wiener process36
4Multidimensional Wiener processes. Markov times43
5Strong Markov property of the Wiener process46
6Martingale properties of the Wiener process50
7Burkholder-Davis-Gundy inequalities and Wald identities for the Wiener process54
8The Wiener process and the heat equation. Martingales57
9Some applications of Theorem 8.664
10The Wiener process and the Laplace operator71
Ch. IIIIto's Stochastic Integral77
1Integral with respect to a Random Orthogonal Measure78
2Ito's stochastic integral with respect to a Wiener process84
3Ito's stochastic integral with variable upper limit90
4Extending the set of Ito-integrable functions. The notion of a local martingale95
5Quadratic variation of stochastic integral and pseudopredictable functions101
6Passage to a limit within the Ito stochastic integral. Ito's inequalities. Convergence in probability105
7Ito's integral with respect to a multidimensional Wiener process113
8Ito's formula115
9Martingale version of Ito's formula. Levy's theorem122
10Stochastic integral with respect to an admissible local martingale127
11Regularly measurable processes134
Ch. IVSome Applications of Ito's Formula141
1Transformation of Ito's formula; particular cases141
2Random time change in stochastic integrals147
3Girsanov's theorem153
4Burkholder-Davis-Gundy inequalities for multidimensional random processes160
Ch. VIto's Stochastic Equations165
1Existence and uniqueness of solutions of Ito stochastic equations166
2Two examples of application of Ito stochastic equations173
3Equations solvable by Euler's method178
4Some properties of Euler's approximations182
5Strong Markov property of solutions of stochastic equations189
6The Kolmogorov equations195
7Derivation of the Kolmogorov equation in the inhomogeneous case200
8Derivation of the Kolmogorov equations in the homogeneous case206
9Probabilistic solutions of partial differential equations212
10Proof of Theorem 9.4215
Ch. VIFurther Methods for Investigating the Smoothness of Probabilistic Solutions of Differential Equations221
1Some generalizations of Theorems V.8.1 and V.8.5222
2Quasiderivatives of solutions of stochastic equations228
3Proofs of Lemmas 1.3 and 1.8232
4Some ideas from the theory of conditional processes236
5A method for investigating the function (V.6.13) for [actual symbol not reproducible]246
Appendix A. Proof of Lemma II.2.4257
Appendix B. Proof of Theorem II.8.1259
List of Notations261
Comments263
References267
Index269

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