Entropies and Fractionality: Entropy Functionals, Small Deviations and Related Integral Equations
暫譯: 熵與分數性：熵函數、小偏差及相關的積分方程
Mishura, Yuliya, Ralchenko, Kostiantyn
- 出版商: CRC
- 出版日期: 2025-10-20
- 售價: $8,460
- 貴賓價: 9.5 折 $8,037
- 語言: 英文
- 頁數: 270
- 裝訂: Hardcover - also called cloth, retail trade, or trade
- ISBN: 1041074786
- ISBN-13: 9781041074786
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相關分類:
機率統計學 Probability-and-statistics
海外代購書籍(需單獨結帳)
商品描述
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, Entropic Value-at-Risk (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 have been 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
商品描述(中文翻譯)
《熵與分數性：熵泛函、小偏差及相關的積分方程》首先系統化並計算了選定的絕對連續機率分佈的各種熵（Shannon 熵、Rényi 熵及其他熵）。接著分析了這些熵的性質。隨後，書中探討了一個相關問題：熵風險度量的計算與性質研究，即熵風險值（Entropic Value-at-Risk, EVaR）。
接下來，書中計算並比較了各種分數高斯過程的一維分佈的熵值。特別關注於分數高斯噪聲，作者對 Shannon 熵的性質及漸近行為進行了詳細分析。此外，還引入了兩個更適合進行分析研究的替代熵泛函。
進一步地，考慮了兩個獨立的帶漂移的 Wiener 過程之和的相對熵泛函，並探討了其最小化和最大化問題。對於帶漂移的混合分數布朗運動（即 Wiener 過程與分數布朗運動之和），也提出了類似的問題。熵最小化問題被簡化為第二類 Fredholm 積分方程，並對其唯一可解性進行了深入研究。
在書的最後部分，研究了帶趨勢的混合分數布朗運動的小偏差優化問題。這個問題與相對熵泛函的最小化密切相關，並使用類似的技術和結果來解決，導致相同類別的積分方程。由於解這類方程因核中存在額外的奇異性而變得具有挑戰性，因此已開發出有效的數值方法來應對這一困難。
特點：
- 對於對熵相關問題感興趣的數學家以及實務工作者，特別是物理、金融和信息理論的專家都非常有用
- 提供了大量的例子和應用，並附有嚴謹的證明
作者簡介
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.
作者簡介(中文翻譯)
尤莉亞·米舒拉於1978年在基輔大學獲得概率與統計的博士學位，並於1990年完成概率與統計的博士後學位（Habilitation）。她目前是基輔塔拉斯·舍甫琴科國立大學概率、統計與精算數學系的教授。她擁有廣泛而多樣的科學興趣，是320多篇研究論文和20多本書籍的作者或合著者。她的研究興趣包括隨機過程的理論與統計、隨機微分方程、分數微積分與分數過程、隨機分析、函數極限定理、概率分佈與隨機系統的熵、金融數學及其他隨機應用。她是多個國際大會和會議的受邀演講者，並組織了一系列會議。她是期刊《概率論與數學統計理論》的主編，並擔任期刊《現代隨機學：理論與應用》的共同主編。她是多個國際研究項目的團隊負責人和參與者。
科斯蒂安丁·拉爾琴科於2012年在基輔塔拉斯·舍甫琴科國立大學獲得概率與統計的博士學位，並於2019年在同一領域完成博士後資格（Habilitation）。他目前擔任基輔塔拉斯·舍甫琴科國立大學概率、統計與精算數學系的教授。他是60多篇研究論文和4本科學專著的作者或合著者。他的研究興趣包括隨機過程的理論與統計分析、分數與多分數過程、常微分與偏微分隨機微分方程、概率分佈與隨機系統的熵度量，以及金融數學。
