This book is devoted to the study of stochastic measures (SMs). An SM is a sigma-additive in probability random function, defined on a sigma-algebra of sets. SMs can be generated by the increments of random processes from many important classes such as square-integrable martingales and fractional Brownian motion, as well as alpha-stable processes. SMs include many well-known stochastic integrators as partial cases.
General Stochastic Measures provides a comprehensive theoretical overview of SMs, including the basic properties of the integrals of real functions with respect to SMs. A number of results concerning the Besov regularity of SMs are presented, along with equations driven by SMs, types of solution approximation and the averaging principle. Integrals in the Hilbert space and symmetric integrals of random functions are also addressed.
The results from this book are applicable to a wide range of stochastic processes, making it a useful reference text for researchers and postgraduate or postdoctoral students who specialize in stochastic analysis.
Author(s): Vadym M. Radchenko
Series: Mathematics and Statistics
Publisher: Wiley-ISTE
Year: 2022
Language: English
Pages: 254
City: London
Cover
Half-Title Page
Title Page
Copyright Page
Contents
Abbreviations and Notations
Introduction
Chapter 1. Integration with Respect to Stochastic Measures
1.1. Preliminaries
1.2. Stochastic measures
1.2.1. Definition and examples of SMs
1.2.2. Convergence defined by an SM
1.3. Integration of deterministic functions
1.4. Limit theorems for integral of deterministic functions
1.4.1. Convergence of ∫A fn dμ
1.4.2. Convergence of ∫X f dμn
1.5. s-finite stochastic measures
1.6. Riemann integral of a random function w.r.t. a deterministic measure
1.6.1. Definition of the integral
1.6.2. Interchange of the order of integration
1.6.3. Iterated integral and integration by parts
1.7. Exercises
1.8. Bibliographical notes
Chapter 2. Path Properties of Stochastic Measures
2.1. Sample functions of stochastic measures and Besov spaces spaces
2.1.1. Besov
2.1.2. Auxiliary lemmas
2.1.3. Stochastic measures on [0, 1]
2.1.4. Stochastic measures on [0, 1]d
2.2. Fourier series expansion of stochastic measures
2.2.1. Convergence of Fourier series of the process µ(t)
2.2.2. Convergence of stochastic integrals
2.3. Continuity of the integral
2.3.1. Estimate of an integral
2.3.2. Parameter dependent integral
2.3.3. Continuity with respect to the upper limit
2.4. Exercises
2.5. Bibliographical notes
Chapter 3. Equations Driven by Stochastic Measures
3.1. Parabolic equation in R (case dµs(x))
3.1.1. Problem and the main result
3.1.2. Lemma about the Hölder continuity in x
3.1.3. Lemma about the Hölder continuity in t
3.2. Heat equation in Rd (case dµ(t))
3.2.1. Additional estimate of an integral
3.2.2. Problem and the main result
3.2.3. Lemma about the Hölder continuity in x
3.2.4. Lemma about the Hölder continuity in t
3.3. Wave equation in R (case dµ(x))
3.3.1. Problem and the main result
3.3.2. Lemma about the Hölder continuity in x
3.3.3. Lemma about the Hölder continuity in t
3.4. Wave equation in R (case dµ(t))
3.4.1. Problem and the main result
3.4.2. Lemma about the Lipschiz continuity in x
3.4.3. Lemma about the Hölder continuity in t
3.5. Parabolic evolution equation in Rd (weak solution, case dµ(t))
3.6. Exercises
3.7. Bibliographical notes
Chapter 4. Approximation of Solutions of the Equations
4.1. Parabolic equation in R (case dµ(x))
4.1.1. Problem and the main result
4.1.2. Auxiliary lemmas
4.1.3. Examples
4.2. Heat equation in Rd (case dµ(t))
4.2.1. Problem and the main result
4.2.2. Auxiliary lemma
4.2.3. Examples
4.3. Wave equation in R (case dµ(t))
4.3.1. Approximation by using the convergence of paths of SMs
4.3.2. Approximation by using the Fourier partial sums
4.3.3. Approximation by using the Fejèr sums
4.3.4. Auxiliary lemma
4.3.5. Example
4.4. Exercises
4.5. Bibliographical notes
Chapter 5. Integration and Evolution Equations in Hilbert Spaces
5.1. Preliminaries
5.2. Equations and integral with a real-valued SM
5.2.1. Integral w.r.t. a real-valued SM
5.2.2. Evolution equations driven by a real-valued SM
5.3. Equations and integrals with a Hilbert space-valued SM
5.3.1. Integrals w.r.t. a U-valued SM
5.3.2. Evolution equations driven by a U-valued SM
5.4. Exercises
5.5. Bibliographical notes
Chapter 6. Symmetric Integrals
6.1. Introduction
6.2. SM has finite strong cubic variation
6.3. Stratonovich-type integral
6.4. SDE driven by an SM
6.5. Wong–Zakai approximation
6.6. Some counterexamples
6.7. Exercises
6.8. Bibliographical notes
Chapter 7. Averaging Principle
7.1. Heat equation
7.1.1. Introduction
7.1.2. The problem
7.1.3. Averaging principle
7.2. Equation with the symmetric integral
7.2.1. Introduction
7.2.2. Averaging principle
7.3. Exercises
7.4. Bibliographical notes
Chapter 8. Solutions to Exercises
References
Index
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