Entropies and Fractionality: Entropy Functionals, Small Deviations and Related Integral Equations starts with a systematization and calculation of various entropies (Shannon, Rényi and some others) of selected absolutely continuous probability distributions. The properties of the entropies are analyzed. Subsequently, a related problem is addressed: the computation and investigation of the properties of the entropic risk measure, EVaR.
Next, the book computes and compares entropy values for the one-dimensional distributions of various fractional Gaussian processes. Special attention is then given to fractional Gaussian noise, where the authors conduct a detailed analysis of the properties and asymptotic behavior of Shannon entropy. Additionally, two alternative entropy functionals are introduced which are more suitable for analytical investigation.
Furthermore, relative entropy functionals for the sum of two independent Wiener processes with drift are considered, and their minimization and maximization are explored. A similar problem is addressed for a mixed fractional Brownian motion (i.e., the sum of a Wiener process and a fractional Brownian motion) with drift. The entropy minimization problem is reduced to a Fredholm integral equation of the second kind, and its unique solvability is thoroughly investigated.
In the final part of the book, the optimization of small deviations for mixed fractional Brownian motion with trend is studied. This problem is closely related to the minimization of relative entropy functionals and is solved using similar techniques and results, which leads to the same class of integral equations. Since solving such equations is challenging due to the presence of an additional singularity in the kernel, efficient numerical methods are developed to address this difficulty.
Features
Useful both for mathematicians interested in problems related to entropy, and for practitioners, especially specialists in physics, finance, and information theory Numerous examples and applications are provided, with rigorous proofs
Preface
Introduction
I Entropies and Entropy Risk Measures Related to Probability Distributions
1 Entropies of Selected Distributions and Their Properties
2 Entropic Risk Measure EVaR in Relation to Selected Distributions
II Entropies and Entropy Functionals Related to Fractional Stochastic
Processes
3 Entropies of Fractional Processes
4 Fractional Gaussian Noise: Entropy, Entropy Rate and Alternative Entropy
Functionals
5 Evaluation of Extreme Values of the Relative Entropy Functionals Related to
the Mixture of Wiener Processes
6 Entropy Optimization for a Mixture of Standard and Fractional Brownian
Motions
III Small Deviations of the Mixed Fractional Processes and Numerical Solution
of the Related Integral Equations
7 Optimization of Small Deviation for Mixed Fractional Brownian Motion with
Trend
8 Approximate Solution of the Integral Equations Involving Kernel with
Additional Singularity
A Elements of Calculus, Fractional Calculus and Stochastics
Bibliography
Index
Yuliya Mishura received her PhD in probability and statistics in Kyiv University in 1978 and completed her postdoctoral degree in probability and statistics (Habilitation) in 1990. She is currently a Professor of the Department of Probability, Statistics and Actuarial Mathematics at Taras Shevchenko National University of Kyiv. Having broad and varied scientific interests, she is the author/coauthor of more than 320 research papers and more than 20 books. Her research interests include theory and statistics of stochastic processes, stochastic differential equations, fractional calculus and fractional processes, stochastic analysis, functional limit theorems, entropies of probability distributions and stochastic systems, financial mathematics and other applications of stochastics. Invited speaker of many international congresses and conferences, organizer of series of conferences. Editor- in-chief of the journal Theory of Probability and Mathematical Statistics, coeditor-in-chief of the journal Modern Stochastics: Theory and Applications. Team leader and participant of many international research projects.
Kostiantyn Ralchenko obtained his PhD in Probability and Statistics from Taras Shevchenko National University of Kyiv in 2012 and completed his postdoctoral qualification (Habilitation) in the same field in 2019. He currently holds the position of Professor in the Department of Probability, Statistics, and Actuarial Mathematics at Taras Shevchenko National University of Kyiv. He is the author/co-author of more than 60 research papers and 4 scientific monographs. His research interests include the theory and statistical analysis of stochastic processes, fractional and multifractional processes, ordinary and partial stochastic differential equations, entropy measures of probability distributions and stochastic systems, as well as financial mathematics.
