Stochastic Dynamical Systems
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Stochastic Dynamical Systems
Dieser einzigartige Band führt den Leser in die mathematische Begriffsbildung für komplexe Systeme ein. Er ist ideal für Studenten der Mathematik, Physik, Chemie und Medizin, die sich in ihrem Studium erstmals mit stochastischen dynamischen Systemen beschäftigen. Das Buch stellt praktische Methoden zur Verfügung, um mit solchen Systemen umgehen zu können, und stellt die zugundeliegenden Definitionen und theoretischen Annahmen, wo erforderlich, klar heraus. Im Gegensatz zu anderen Büchern über dieses Gebiet, die oft einen bestimmten Zugang bevorzugen, deckt Stochastical Dynamical Systems eine Vielzahl von stochastischen und statistischen Methoden ab, die für die Untersuchung von komplexen Systemen wie Polymerschmelzen, dem menschlichen Körper und der Atmosphäre absolut notwendig sind. Das Buch behandelt die Datenanalyse ebenso wie Simulationsmethoden für gegebene Modelle. Die ganze Vielfalt der klassischen und neuartigen Begriffe der mathematischen Stochastik wird in einem leicht verständlichen Stil erklärt, so daß die Leser diese Konzepte leicht für die Untersuchung ihrer Daten anwenden können.
Random Dynamical Systems
Author: Ludwig Arnold
language: en
Publisher: Springer Science & Business Media
Release Date: 2013-04-17
Background and Scope of the Book This book continues, extends, and unites various developments in the intersection of probability theory and dynamical systems. I will briefly outline the background of the book, thus placing it in a systematic and historical context and tradition. Roughly speaking, a random dynamical system is a combination of a measure-preserving dynamical system in the sense of ergodic theory, (D,F,lP', (B(t))tE'lf), 'II'= JR+, IR, z+, Z, with a smooth (or topological) dy namical system, typically generated by a differential or difference equation :i: = f(x) or Xn+l = tp(x.,), to a random differential equation :i: = f(B(t)w,x) or random difference equation Xn+l = tp(B(n)w, Xn)· Both components have been very well investigated separately. However, a symbiosis of them leads to a new research program which has only partly been carried out. As we will see, it also leads to new problems which do not emerge if one only looks at ergodic theory and smooth or topological dynam ics separately. From a dynamical systems point of view this book just deals with those dynamical systems that have a measure-preserving dynamical system as a factor (or, the other way around, are extensions of such a factor). As there is an invariant measure on the factor, ergodic theory is always involved.
Random Dynamical Systems in Finance
The theory and applications of random dynamical systems (RDS) are at the cutting edge of research in mathematics and economics, particularly in modeling the long-run evolution of economic systems subject to exogenous random shocks. Despite this interest, there are no books available that solely focus on RDS in finance and economics. Exploring this emerging area, Random Dynamical Systems in Finance shows how to model RDS in financial applications. Through numerous examples, the book explains how the theory of RDS can describe the asymptotic and qualitative behavior of systems of random and stochastic differential/difference equations in terms of stability, invariant manifolds, and attractors. The authors present many models of RDS and develop techniques for implementing RDS as approximations to financial models and option pricing formulas. For example, they approximate geometric Markov renewal processes in ergodic, merged, double-averaged, diffusion, normal deviation, and Poisson cases and apply the obtained results to option pricing formulas. With references at the end of each chapter, this book provides a variety of RDS for approximating financial models, presents numerous option pricing formulas for these models, and studies the stability and optimal control of RDS. The book is useful for researchers, academics, and graduate students in RDS and mathematical finance as well as practitioners working in the financial industry.