General Stochastic Measures: Integration, Path Properties and Equations

ISBN-13: 9781786308283 / Angielski / Twarda / 2022 / 272 str.

Radchenko

Spis treści

Abbreviations and Notations ixIntroduction xiChapter 1 Integration with Respect to Stochastic Measures 11.1. Preliminaries 11.2. Stochastic measures 21.2.1. Definition and examples of SMs 21.2.2. Convergence defined by an SM 51.3. Integration of deterministic functions 61.4. Limit theorems for integral of deterministic functions 111.4.1 Convergence of integral A fn dmu 131.4.2 Convergence of integral X fdmun 141.5. sigma-finite stochastic measures 161.6. Riemann integral of a random function w.r.t. a deterministic measure 211.6.1. Definition of the integral 211.6.2. Interchange of the order of integration 271.6.3. Iterated integral and integration by parts 291.7. Exercises 321.8. Bibliographical notes 34Chapter 2 Path Properties of Stochastic Measures 352.1. Sample functions of stochastic measures and Besov spaces 352.1.1. Besov spaces 352.1.2. Auxiliary lemmas 372.1.3 Stochastic measures on [0, 1] 422.1.4 Stochastic measures on [0, 1] d 442.2. Fourier series expansion of stochastic measures 462.2.1 Convergence of Fourier series of the process mu(t) 462.2.2. Convergence of stochastic integrals 492.3. Continuity of the integral 512.3.1. Estimate of an integral 512.3.2. Parameter dependent integral 542.3.3. Continuity with respect to the upper limit 552.4. Exercises 572.5. Bibliographical notes 59Chapter 3 Equations Driven by Stochastic Measures 613.1 Parabolic equation in R (case dmu¯sigma (x)) 613.1.1. Problem and the main result 613.1.2 Lemma About the Hölder Continuity in X 703.1.3 Lemma about the Hölder continuity in t 753.2 Heat equation in R¯d (case dmu(t)) 783.2.1. Additional estimate of an integral 783.2.2. Problem and the main result 803.2.3 Lemma About the Hölder Continuity in X 843.2.4 Lemma about the Hölder continuity in t 913.3 Wave equation in R (case dmu(x)) 993.3.1. Problem and the main result 993.3.2 Lemma About the Hölder Continuity in X 1023.3.3 Lemma about the Hölder continuity in t 1063.4 Wave equation in R (case dmu(t)) 1083.4.1. Problem and the main result 1083.4.2 Lemma About the Lipschiz Continuity in X 1093.4.3 Lemma about the Hölder continuity in t 1113.5 Parabolic evolution equation in R d (weak solution, case dmu(t)) 1143.6. Exercises 1193.7. Bibliographical notes 120Chapter 4 Approximation of Solutions of the Equations 1234.1 Parabolic equation in R (case dmu(x)) 1234.1.1. Problem and the main result 1234.1.2. Auxiliary lemmas 1304.1.3. Examples 1334.2 Heat equation in R¯d (case dmu(t)) 1354.2.1. Problem and the main result 1354.2.2. Auxiliary lemma 1384.2.3. Examples 1394.3 Wave equation in R (case dmu(t)) 1404.3.1. Approximation by using the convergence of paths of SMs 1404.3.2. Approximation by using the Fourier partial sums 1424.3.3. Approximation by using the Fejèr sums 1494.3.4. Auxiliary lemma 1514.3.5. Example 1534.4. Exercises 1544.5. Bibliographical notes 155Chapter 5. Integration and Evolution Equations in Hilbert Spaces 1575.1. Preliminaries 1575.2. Equations and integral with a real-valued SM 1605.2.1. Integral w.r.t. a real-valued SM 1605.2.2. Evolution equations driven by a real-valued SM 1635.3. Equations and integrals with a Hilbert space-valued SM 1665.3.1. Integrals w.r.t. a U-valued SM 1665.3.2. Evolution equations driven by a U-valued SM 1715.4. Exercises 1725.5. Bibliographical notes 173Chapter 6 Symmetric Integrals 1756.1. Introduction 1756.2. SM has finite strong cubic variation 1766.3. Stratonovich-type integral 1776.4. SDE driven by an SM 1816.5. Wong-Zakai approximation 1836.6. Some counterexamples 1886.7. Exercises 1926.8. Bibliographical notes 193Chapter 7 Averaging Principle 1957.1. Heat equation 1957.1.1. Introduction 1957.1.2. The problem 1967.1.3. Averaging principle 1977.2. Equation with the symmetric integral 2027.2.1. Introduction 2027.2.2. Averaging principle 2057.3. Exercises 2117.4. Bibliographical notes 212Chapter 8 Solutions to Exercises 213References 231Index 241

O autorze

Vadym M. Radchenko is Full Professor in the Department of Mathematical Analysis at Taras Shevchenko National University of Kyiv, Ukraine. His research interests include stochastic integration and stochastic partial differential equations.