انتگرال‌های تصادفی (مونوگرافی آمار و آمار ریاضی)

The AMS is excited to bring this volume, originally published in 1969, back into print. This well-written book has been used for many years to learn about stochastic integrals. The author starts with the presentation of Brownian motion, then deals with stochastic integrals and differentials, including the famous Itô lemma. The rest of the book is devoted to various topics of stochastic integral equations and stochastic integral equations on smooth manifolds. E. B. Dynkin wrote about the original edition in Mathematical Reviews: "This little book is a brilliant introduction to an important boundary field between the theory of probability and differential equations." These words continue to ring true today. This classic book is ideal for supplementary reading or independent study. It is suitable for graduate students and researchers interested in probability, stochastic processes, and their applications. Title......Page 1 Copyrigth Page......Page 2 Preface......Page 5 Contents......Page 7 List of Notations......Page 9 Introduction......Page 15 1.1 Gaussian Families......Page 17 1.2 Construction of the Brownian Motion......Page 19 1.3 Simplest Properties of the Brownian Motion......Page 23 1.4 A Martingale Inequality......Page 25 1.5 The Law of the Iterated Logarithm......Page 26 1.6 Levy's Modulus......Page 28 1.7 Several-Dimensional Brownian Motion......Page 31 2.1 Wiener's Definition of the Stochastic Integral......Page 34 2.2 Ito's Definition of the Stochastic Integral......Page 35 2.3 Simplest Properties of the Stochastic Integral......Page 38 2.4 Computation of a Stochastic Integral......Page 42 2.5 A Time Substitution......Page 43 2.6 Stochastic Differentials and Ito's Lemma......Page 46 2.7 Solution of the Simplest Stochastic Differential Equation......Page 49 2.8 Stochastic Differentials under a Time Substitution......Page 55 2.9 Stochastic Integrals and Differentials for Several-Dimensional Brownian Motion......Page 57 3.1 Diffusions......Page 64 3.2 Solution of dx=e(x) db + f(x)dt for Coefficients with Bounded Slope......Page 66 3.3 Solution of dx=e(x) db + f(x)dt for General Coefficients Belonging to C_1(R^1)......Page 68 3.4 Lamperti's Method......Page 74 3.5 Forward Equation......Page 75 3.6 Feller's Test for Explosions......Page 79 3.7 Cameron-Martin's Formula......Page 81 3.8 Brownian Local Time......Page 82 3.9 Reflecting Barriers......Page 85 3.10 Some Singular Equations......Page 91 4.1 Manifolds and Elliptic Operators......Page 96 4.2 Weyl's Lemma......Page 99 4.3 Diffusions on a Manifold......Page 104 4.4 Explosions and Harmonic Functions......Page 112 4.5 Hasminskii's Test for Explosions......Page 116 4.6 Covering Brownian Motions......Page 122 4.7 Brownian Motions on a Lie Group......Page 129 4.8 Injection......Page 131 4.9 Brownian Motion of Symmetric Matrices......Page 137 4.10 Brownian Motion with Oblique Reflection......Page 140 References......Page 147 Subject Index......Page 153