General Stochastic Measures

Abbreviations and Notations ix Introduction xi Chapter 1 Integration with Respect to Stochastic Measures 1 1.1. Preliminaries 1 1.2. Stochastic measures 2 1.2.1. Definition and examples of SMs 2 1.2.2. Convergence defined by an SM 5 1.3. Integration of deterministic functions 6 1.4. Limit theorems for integral of deterministic functions 11 1.4.1 Convergence of ∫A fn dμ 13 1.4.2 Convergence of ∫X  fdμn 14 1.5. σ-finite stochastic measures 16 1.6. Riemann integral of a random function w.r.t. a deterministic measure 21 1.6.1. Definition of the integral 21 1.6.2. Interchange of the order of integration 27 1.6.3. Iterated integral and integration by parts 29 1.7. Exercises 32 1.8. Bibliographical notes 34 Chapter 2 Path Properties of Stochastic Measures 35 2.1. Sample functions of stochastic measures and Besov spaces 35 2.1.1. Besov spaces 35 2.1.2. Auxiliary lemmas 37 2.1.3 Stochastic measures on [0, 1] 42 2.1.4 Stochastic measures on [0, 1] d 44 2.2. Fourier series expansion of stochastic measures 46 2.2.1 Convergence of Fourier series of the process μ(t) 46 2.2.2. Convergence of stochastic integrals 49 2.3. Continuity of the integral 51 2.3.1. Estimate of an integral 51 2.3.2. Parameter dependent integral 54 2.3.3. Continuity with respect to the upper limit 55 2.4. Exercises 57 2.5. Bibliographical notes 59 Chapter 3 Equations Driven by Stochastic Measures 61 3.1 Parabolic equation in R (case dμσ (x)) 61 3.1.1. Problem and the main result 61 3.1.2 Lemma About the Hölder Continuity in X 70 3.1.3 Lemma about the Hölder continuity in t 75 3.2 Heat equation in Rd (case dμ(t)) 78 3.2.1. Additional estimate of an integral 78 3.2.2. Problem and the main result 80 3.2.3 Lemma About the Hölder Continuity in X 84 3.2.4 Lemma about the Hölder continuity in t 91 3.3 Wave equation in R (case dμ(x)) 99 3.3.1. Problem and the main result 99 3.3.2 Lemma About the Hölder Continuity in X 102 3.3.3 Lemma about the Hölder continuity in t 106 3.4 Wave equation in R (case dμ(t)) 108 3.4.1. Problem and the main result 108 3.4.2 Lemma About the Lipschiz Continuity in X 109 3.4.3 Lemma about the Hölder continuity in t 111 3.5 Parabolic evolution equation in R d (weak solution, case dμ(t)) 114 3.6. Exercises 119 3.7. Bibliographical notes 120 Chapter 4 Approximation of Solutions of the Equations 123 4.1 Parabolic equation in R (case dμ(x))  123 4.1.1. Problem and the main result 123 4.1.2. Auxiliary lemmas 130 4.1.3. Examples 133 4.2 Heat equation in Rd (case dμ(t)) 135 4.2.1. Problem and the main result 135 4.2.2. Auxiliary lemma 138 4.2.3. Examples 139 4.3 Wave equation in R (case dμ(t)) 140 4.3.1. Approximation by using the convergence of paths of SMs 140 4.3.2. Approximation by using the Fourier partial sums 142 4.3.3. Approximation by using the Fejèr sums 149 4.3.4. Auxiliary lemma 151 4.3.5. Example 153 4.4. Exercises 154 4.5. Bibliographical notes 155 Chapter 5. Integration and Evolution Equations in Hilbert Spaces 157 5.1. Preliminaries 157 5.2. Equations and integral with a real-valued SM 160 5.2.1. Integral w.r.t. a real-valued SM 160 5.2.2. Evolution equations driven by a real-valued SM 163 5.3. Equations and integrals with a Hilbert space-valued SM 166 5.3.1. Integrals w.r.t. a U-valued SM 166 5.3.2. Evolution equations driven by a U-valued SM 171 5.4. Exercises 172 5.5. Bibliographical notes 173 Chapter 6 Symmetric Integrals 175 6.1. Introduction 175 6.2. SM has finite strong cubic variation 176 6.3. Stratonovich-type integral 177 6.4. SDE driven by an SM 181 6.5. Wong–Zakai approximation 183 6.6. Some counterexamples 188 6.7. Exercises 192 6.8. Bibliographical notes 193 Chapter 7 Averaging Principle 195 7.1. Heat equation 195 7.1.1. Introduction 195 7.1.2. The problem 196 7.1.3. Averaging principle 197 7.2. Equation with the symmetric integral 202 7.2.1. Introduction 202 7.2.2. Averaging principle 205 7.3. Exercises 211 7.4. Bibliographical notes 212 Chapter 8 Solutions to Exercises 213 References 231 Index 241