Yu. V. Prokhorov A. N. Shiryaev Probability Theory III Stochastic Calculus Springer Stochastic Calculus S.V. Anulova, A.Yu. Veretennikov, N.V. Krylov, R.Sh. Liptser, A.N. Shiryaev Translated from the Russian by P.B. Slater Contents Preface 5 Chapter 1. Introduction to Stochastic Calculus (N.V. Krylov) 7 §1. Brownian Motion and the Wiener Process 7 §2. Probabilistic Construction of the Solution of the Heat Equation. Relation Between the Wiener Process and the Laplace Operator . 15 §3. The Ito Integral and Differentiation Rules for Composite Stochastic Functions 19 §4. Stochastic Differential Equations and Diffusion Processes. Girsanov's Theorem 26 §5. Stochastic Differential Equations with Boundary Conditions 32 References 36 Chapter 2. Stochastic Differential and Evolution Equations 38 I. Stochastic Differential Equations (SDEs) (S.V. Anulova, A.Yu. Veretennikov) 38 §1. Strong Solutions of Stochastic Differential Equations 38 §2. Weak Solutions of Stochastic Differential Equations with Non-Smooth Coefficients in Ed 50 §3. Differentiation of Solutions of SDEs with Respect to the Initial Data 54 2 Stochastic Calculus §4. Invariant Measure of a Diffusion Process 57 §5. Support of a Diffusion 59 §6. Stochastic Differential Equations in Domains 63 References i 72 II. Stochastic Evolution Equations (A.Yu. Veretennikov) 76 §1. Introduction 76 §2. Martingales and Stochastic Integrals in Hilbert Space 77 §3. The Ito Formula for the Square of the Norm 81 §4. Stochastic Differential Equations of Monotone Type in Banach Spaces 82 §5. Stochastic Partial Differential Equations I. The First Boundary-Value Problem for Parabolic-Type Non-Linear Equations 85 §6. Stochastic Partial Differential Equations II. The Cauchy Problem for Second-Order Linear Equations 88 References 90 III. Stochastic Calculus (Malliavin Calculus). Applications to Stochastic Differential Equations (A.Yu. Veretennikov) 91 §1. Introduction 91 §2. Stochastic Derivatives 92 §3. Rules of the Malliavin Calculus 96 §4. Smoothness of the Density (Scheme of the Proof) 98 §5. The Bismut Approach 1 99 §6. The Bismut Approach 2. Stochastic Differential Equations 101 §7. Stochastic Differential Equations (Smoothness of the Density with Respect to Inverse Variables) 107 References 108 Chapter 3. Stochastic Calculus on Filtered Probability Spaces (R.Sh. Liptser, A.N. Shiryaev) Ill I. Elements of the General Theory of Stochastic Processes Ill §1. Kolmogorov's Axioms and the Stochastic Basis Ill §2. Stopping Times, Adapted Stochastic Processes, Optional and Predictable c-Algebras. Classification of Stopping Times 112 §3. Martingales and Local Martingales 116 §4. Increasing Processes. Doob-Meyer Decomposition. Compensators . 118 §5. Random Measures. Integral Random Measures 120 §6. Locally Square-Integrable Martingales. The Quadratic Characteristic 123 §7. Decomposition of Local Martingales " 123 Contents 3 II. Semimartingales. Stochastic Integrals 125 §1. Semimartingales. Quadratic Variation. Quasimartingales 125 §2. Construction of Stochastic Integrals with Respect to Semimartingales 127 §3. The Ito Formula 129 §4. Construction of Stochastic Integrals with Respect to Random Measures 131 §5. Characteristics of Semimartingales. The Triple of Predictable Characteristics T = (B,C,v). Martingale and Semimartingale Problems. Examples 133 §6. Integral Representation of Local Martingales 137 §7. Stability of the Class of Semimartingales with Respect to a Series of Transformations 138 III. Absolute Continuity and Singularity of Probability Distributions . 140 §1. Local Density. Lebesgue Decomposition 140 §2. Girsanov's Theorem and its Generalization. Transformation of Predictable Characteristics 141 §3. The Hellinger Integral and the Hellinger Process 143 §4. General and Predictable Criteria of Absolute Continuity and Singularity of Probability Measures 147 §5. Particular Cases 148 Commentary to Chapter 3 152 References 154 Chapter 4. Martingales and Limit Theorems for Stochastic Processes (R.Sh. Liptser, A.N. Shiryaev) 158 I. Theory: Weak Convergence of Probability Measures on Metric Spaces 158 §1. Introduction 158 §2. Different Types of Convergence. Skorokhod Topology 160 §3. Brief Review of a Number of Classical Limit Theorems of Probability Theory 166 §4. Convergence of Processes with Independent Increments 178 §5. The Convergence of Semimartingales to Processes with Independent Increments 188 §6. Relative Compactness and Tightness of Families of Distributions of Semimartingales 200 §7. Convergence of Semimartingales to a Semimartingale 202 §8. The Martingale Problem 210 4 Stochastic Calculus II. Applications: The Invariance Principle and Diffusion Approximation 213 §1. The Invariance Principle for Stationary and Markov Processes ... 213 §2. The Stochastic Averaging Principle in Models without Diffusion . 226 §3. Diffusion Approximation of Semimartingales. The Averaging Principle in Models with Diffusion 229 §4. Diffusion Approximation for Systems with Physical White Noise . 233 §5. Diffusion Approximation for Semimartingales with Normal Reflection in a Convex Domain 237 Commentary to Chapter 4 243 References 244 Author Index 249 Subject Index 251.