Thomas Mikosch

This book deals with extreme value theory for univariate and multivariate time series models characterized by power-law tails. These include the classical ARMA models with heavy-tailed noise and financial econometrics models such as the GARCH and stochastic volatility models. Rigorous descriptions of power-law tails are provided through the concept of regular variation. Several chapters are devoted to the exploration of regularly varying structures. The remaining chapters focus on the impact of heavy tails on time series, including the study of extremal cluster phenomena through point process techniques. A major part of the book investigates how extremal dependence alters the limit structure of sample means, maxima, order statistics, sample autocorrelations. This text illuminates the theory through hundreds of examples and as many graphs showcasing its applications to real-life financial and simulated data. The book can serve as a text for PhD and Master courses on applied probability, extreme value theory, and time series analysis. It is a unique reference source for the heavy-tail modeler. Its reference quality is enhanced by an exhaustive bibliography, annotated by notes and comments making the book broadly and easily accessible.

This book deals with extreme value theory for univariate and multivariate time series models characterized by power-law tails. These include the classical ARMA models with heavy-tailed noise and financial econometrics models such as the GARCH and stochastic volatility models. Rigorous descriptions of power-law tails are provided through the concept of regular variation. Several chapters are devoted to the exploration of regularly varying structures. The remaining chapters focus on the impact of heavy tails on time series, including the study of extremal cluster phenomena through point process techniques. A major part of the book investigates how extremal dependence alters the limit structure of sample means, maxima, order statistics, sample autocorrelations. This text illuminates the theory through hundreds of examples and as many graphs showcasing its applications to real-life financial and simulated data. The book can serve as a text for PhD and Master courses on applied probability, extreme value theory, and time series analysis. It is a unique reference source for the heavy-tail modeler. Its reference quality is enhanced by an exhaustive bibliography, annotated by notes and comments making the book broadly and easily accessible.

Oriental Jazz Improvisation - Microtonality and Harmony is a comprehensive comparative study that explores the incorporation of Turkish, Arabic, and North Indian scales and modes into jazz improvisation and modulation. The book includes a multitude of scales and modes from these music cultures, along with their transpositions. These scales and modes are essential for any musician interested in South-Western Asian or North Indian music. Besides, the book delves into numerous scales and modes from other cultures' folk music, such as Greek rebetiko, Bulgarian wedding music, or Jewish music, highlighting parallels and similarities among these diverse music traditions. Each scale and mode is presented in its purest - microtonal - form, including the exact interval values in the particular tuning used in the classical Turkish, Arabic, or North Indian music tradition. Additionally, the basics of the respective music theories are imparted. Further, the genuine scales and modes are transcribed into the equal temperament and several practical techniques are shown how to apply them in a jazz context. Also, a variety of useful methods are demonstrated of how South-Eastern European Romani people and Jews use some of those scales and modes in improvisation. It further explores how scales, which may appear identical but exist under different names in various music cultures, are applied differently in these traditions. Scales that can date back to ancient Greece or even as far as Babylonia. This book is going to make you neither a Turkish, Arabic, nor Indian musician; but it surely will broaden your musical vocabulary.

T RK SANAT M ZİĞİ - Die t rkische Kunstmusik, eine der gr ten Kunstmusiken der Welt. Sowohl Perser als auch Araber hinterlie en tiefe Spuren in dieser Musik, die trotz ihres homophonen Charakters, da ihr keine Harmonie im klassischen Sinne innewohnt, einen beeindruckenden harmonischen Reichtum erlangt hat und mit der Expansion des Osmanischen Reiches in S dosteuropa und insbesondere dem Balkan, die Musiktraditionen dieser L nder (z. B. Bulgarien, Griechenland etc.) stark beeinflusst hat. Dieses Buch enth lt ber 50 der g ngigsten Makamlar und ihre Tonleitern sowie deren Modulationen. Au erdem wird ausf hrlich auf das aktuelle t rkische Kommasystem sowie die Geschichte der Tonleitern der t rkischen Kunstmusik eingegangen, deren Wurzeln sich vom Osmanischen Reich ber Arabien und Persien bis ins antike Griechenland zur ckverfolgen lassen. Dar ber hinaus geht es auch auf die Diskrepanzen zwischen der Musik, wie sie in osmanischer Zeit gespielt wurde und der heutigen Notation ein.

In this monograph the authors give a systematic approach to the probabilistic properties of the fixed point equation X=AX+B. A probabilistic study of the stochastic recurrence equation X_t=A_tX_{t-1}+B_t for real- and matrix-valued random variables A_t, where (A_t,B_t) constitute an iid sequence, is provided. The classical theory for these equations, including the existence and uniqueness of a stationary solution, the tail behavior with special emphasis on power law behavior, moments and support, is presented. The authors collect recent asymptotic results on extremes, point processes, partial sums (central limit theory with special emphasis on infinite variance stable limit theory), large deviations, in the univariate and multivariate cases, and they further touch on the related topics of smoothing transforms, regularly varying sequences and random iterative systems. The text gives an introduction to the Kesten-Goldie theory for stochastic recurrence equations of the type X_t=A_tX_{t-1}+B_t. It provides the classical results of Kesten, Goldie, Guivarc'h, and others, and gives an overview of recent results on the topic. It presents the state-of-the-art results in the field of affine stochastic recurrence equations and shows relations with non-affine recursions and multivariate regular variation.

In this monograph the authors give a systematic approach to the probabilistic properties of the fixed point equation X=AX+B. A probabilistic study of the stochastic recurrence equation X_t=A_tX_{t-1}+B_t for real- and matrix-valued random variables A_t, where (A_t,B_t) constitute an iid sequence, is provided. The classical theory for these equations, including the existence and uniqueness of a stationary solution, the tail behavior with special emphasis on power law behavior, moments and support, is presented. The authors collect recent asymptotic results on extremes, point processes, partial sums (central limit theory with special emphasis on infinite variance stable limit theory), large deviations, in the univariate and multivariate cases, and they further touch on the related topics of smoothing transforms, regularly varying sequences and random iterative systems. The text gives an introduction to the Kesten-Goldie theory for stochastic recurrence equations of the type X_t=A_tX_{t-1}+B_t. It provides the classical results of Kesten, Goldie, Guivarc'h, and others, and gives an overview of recent results on the topic. It presents the state-of-the-art results in the field of affine stochastic recurrence equations and shows relations with non-affine recursions and multivariate regular variation.

Both in insurance and in finance applications, questions involving extremal events (such as large insurance claims, large fluctuations in financial data, stock market shocks, risk management, ...) play an increasingly important role. This book sets out to bridge the gap between the existing theory and practical applications both from a probabilistic as well as from a statistical point of view. Whatever new theory is presented is always motivated by relevant real-life examples. The numerous illustrations and examples, and the extensive bibliography make this book an ideal reference text for students, teachers and users in the industry of extremal event methodology.

The second edition of this book contains both basic and more advanced - terial on non-life insurance mathematics. Parts I and II of the book cover the basic course of the 1rst edition; this text has changed very little. It aims at the undergraduate (bachelor) actuarial student as a 1rst introduction to the topics of non-life insurance mathematics. Parts III and IV are new. They can serve as an independent course on stochastic models of non-life insurance mathematics at the graduate (master) level. The basic themes in all parts of this book are point process theory, the Poisson and compound Poisson processes. Point processes constitute an - portant part of modern stochastic process theory. They are well understood models and have applications in a wide range of applied probability areas such as stochastic geometry, extreme value theory, queuing and large computer networks, insurance and finance. The main idea behind a point process is counting. Counting is bread and butter in non-life insurance: the modeling of claim numbers is one of the - jor tasks of the actuary. Part I of this book extensively deals with counting processes on the real line, such as the Poisson, renewal and mixed Poisson processes. These processes can be studied in the point process framework as well, but such an approach requires more advanced theoretical tools.

Modelling with the Itô integral or stochastic differential equations has become increasingly important in various applied fields, including physics, biology, chemistry and finance. However, stochastic calculus is based on a deep mathematical theory.This book is suitable for the reader without a deep mathematical background. It gives an elementary introduction to that area of probability theory, without burdening the reader with a great deal of measure theory. Applications are taken from stochastic finance. In particular, the Black-Scholes option pricing formula is derived. The book can serve as a text for a course on stochastic calculus for non-mathematicians or as elementary reading material for anyone who wants to learn about Itô calculus and/or stochastic finance.

Both in insurance and in finance applications, questions involving extremal events (such as large insurance claims, large fluctuations in financial data, stock market shocks, risk management, ...) play an increasingly important role.