Peter E. Kloeden's Books

Numerical Solution of Stochastic Differential Equations

By: Peter E. Kloeden,Eckhard Platen

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Numerical Solution of Stochastic Differential Equations

By: Peter E. Kloeden,Eckhard Platen

The numerical analysis of stochastic differential equations (SDEs) differs significantly from that of ordinary differential equations. This book provides an easily accessible introduction to SDEs, their applications and the numerical methods to solve such equations. From the reviews: "The authors draw upon their own research and experiences in obviously many disciplines... considerable time has obviously been spent writing this in the simplest language possible." --ZAMP

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Page Count

666

Publisher

Springer Science & Business Media

Published Date

ISBN 10

3662126168

ISBN 13

9783662126165

Chaotic Numerics

An International Workshop on the Approximation and Computation of Complicated Dynamical Behavior, Deakin University, Geelong, Australia, July 12-16, 1993

By: Peter E. Kloeden

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Chaotic Numerics

An International Workshop on the Approximation and Computation of Complicated Dynamical Behavior, Deakin University, Geelong, Australia, July 12-16, 1993

By: Peter E. Kloeden

Much of what is known about specific dynamical systems is obtained from numerical experiments. Although the discretization process usually has no significant effect on the results for simple, well-behaved dynamics, acute sensitivity to changes in initial conditions is a hallmark of chaotic behavior. How confident can one be that the numerical dynamics reflects that of the original system? Do numerically calculated trajectories always shadow a true one? What role does numerical analysis play in the study of dynamical systems? And conversely, can advances in dynamical systems provide new insights into numerical algorithms? These and related issues were the focus of the workshop on Chaotic Numerics, held at Deakin University in Geelong, Australia, in July 1993. The contributions to this book are based on lectures presented during the workshop and provide a broad overview of this area of research.

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Page Count

290

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821851845

ISBN 13

9780821851845

Recent Developments in Computational Finance

Foundations, Algorithms and Applications

By: Thomas Gerstner,Peter E. Kloeden

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Recent Developments in Computational Finance

Foundations, Algorithms and Applications

By: Thomas Gerstner,Peter E. Kloeden

Computational finance is an interdisciplinary field which joins financial mathematics, stochastics, numerics and scientific computing. Its task is to estimate as accurately and efficiently as possible the risks that financial instruments generate. This volume consists of a series of cutting-edge surveys of recent developments in the field written by leading international experts. These make the subject accessible to a wide readership in academia and financial businesses. The book consists of 13 chapters divided into 3 parts: foundations, algorithms and applications. Besides surveys of existing results, the book contains many new previously unpublished results.

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Page Count

481

Publisher

World Scientific

Published Date

ISBN 10

9814436437

ISBN 13

9789814436434

Metric Spaces of Fuzzy Sets

Theory and Applications

By: Phil Diamond,Peter E. Kloeden

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Metric Spaces of Fuzzy Sets

Theory and Applications

By: Phil Diamond,Peter E. Kloeden

The primary aim of the book is to provide a systematic development of the theory of metric spaces of normal, upper semicontinuous fuzzy convex fuzzy sets with compact support sets, mainly on the base space ?n. An additional aim is to sketch selected applications in which these metric space results and methods are essential for a thorough mathematical analysis.This book is distinctly mathematical in its orientation and style, in contrast with many of the other books now available on fuzzy sets, which, although all making use of mathematical formalism to some extent, are essentially motivated by and oriented towards more immediate applications and related practical issues. The reader is assumed to have some previous undergraduate level acquaintance with metric spaces and elementary functional analysis.

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Page Count

192

Publisher

World Scientific

Published Date

ISBN 10

9810217315

ISBN 13

9789810217310

An Introduction to the Numerical Simulation of Stochastic Diﬀerential Equations

By: Desmond J. Higham ,Peter E. Kloeden

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An Introduction to the Numerical Simulation of Stochastic Diﬀerential Equations

By: Desmond J. Higham ,Peter E. Kloeden

This book provides a lively and accessible introduction to the numerical solution of stochastic differential equations with the aim of making this subject available to the widest possible readership. It presents an outline of the underlying convergence and stability theory while avoiding technical details. Key ideas are illustrated with numerous computational examples and computer code is listed at the end of each chapter. The authors include 150 exercises, with solutions available online, and 40 programming tasks. Although introductory, the book covers a range of modern research topics, including Itô versus Stratonovich calculus, implicit methods, stability theory, nonconvergence on nonlinear problems, multilevel Monte Carlo, approximation of double stochastic integrals, and tau leaping for chemical and biochemical reaction networks. An Introduction to the Numerical Simulation of Stochastic Differential Equations is appropriate for undergraduates and postgraduates in mathematics, engineering, physics, chemistry, finance, and related disciplines, as well as researchers in these areas. The material assumes only a competence in algebra and calculus at the level reached by a typical first-year undergraduate mathematics class, and prerequisites are kept to a minimum. Some familiarity with basic concepts from numerical analysis and probability is also desirable but not necessary.

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Page Count

293

Publisher

SIAM

Published Date

ISBN 10

161197643X

ISBN 13

9781611976434

Nonautonomous Dynamical Systems

By: Peter E. Kloeden,Martin Rasmussen

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Nonautonomous Dynamical Systems

By: Peter E. Kloeden,Martin Rasmussen

The theory of nonautonomous dynamical systems in both of its formulations as processes and skew product flows is developed systematically in this book. The focus is on dissipative systems and nonautonomous attractors, in particular the recently introduced concept of pullback attractors. Linearization theory, invariant manifolds, Lyapunov functions, Morse decompositions and bifurcations for nonautonomous systems and set-valued generalizations are also considered as well as applications to numerical approximations, switching systems and synchronization. Parallels with corresponding theories of control and random dynamical systems are briefly sketched. With its clear and systematic exposition, many examples and exercises, as well as its interesting applications, this book can serve as a text at the beginning graduate level. It is also useful for those who wish to begin their own independent research in this rapidly developing area.

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Page Count

274

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821868713

ISBN 13

9780821868713

Random Ordinary Differential Equations and Their Numerical Solution

By: Xiaoying Han,Peter E. Kloeden

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Random Ordinary Differential Equations and Their Numerical Solution

By: Xiaoying Han,Peter E. Kloeden

This book is intended to make recent results on the derivation of higher order numerical schemes for random ordinary differential equations (RODEs) available to a broader readership, and to familiarize readers with RODEs themselves as well as the closely associated theory of random dynamical systems. In addition, it demonstrates how RODEs are being used in the biological sciences, where non-Gaussian and bounded noise are often more realistic than the Gaussian white noise in stochastic differential equations (SODEs). RODEs are used in many important applications and play a fundamental role in the theory of random dynamical systems. They can be analyzed pathwise with deterministic calculus, but require further treatment beyond that of classical ODE theory due to the lack of smoothness in their time variable. Although classical numerical schemes for ODEs can be used pathwise for RODEs, they rarely attain their traditional order since the solutions of RODEs do not have sufficient smoothness to have Taylor expansions in the usual sense. However, Taylor-like expansions can be derived for RODEs using an iterated application of the appropriate chain rule in integral form, and represent the starting point for the systematic derivation of consistent higher order numerical schemes for RODEs. The book is directed at a wide range of readers in applied and computational mathematics and related areas as well as readers who are interested in the applications of mathematical models involving random effects, in particular in the biological sciences.The level of this book is suitable for graduate students in applied mathematics and related areas, computational sciences and systems biology. A basic knowledge of ordinary differential equations and numerical analysis is required.

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Page Count

252

Publisher

Springer

Published Date

ISBN 10

981106265X

ISBN 13

9789811062650

An Introduction To Nonautonomous Dynamical Systems And Their Attractors

By: Peter Kloeden,Meihua Yang

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An Introduction To Nonautonomous Dynamical Systems And Their Attractors

By: Peter Kloeden,Meihua Yang

The nature of time in a nonautonomous dynamical system is very different from that in autonomous systems, which depend only on the time that has elapsed since starting rather than on the actual time itself. Consequently, limiting objects may not exist in actual time as in autonomous systems. New concepts of attractors in nonautonomous dynamical system are thus required.In addition, the definition of a dynamical system itself needs to be generalised to the nonautonomous context. Here two possibilities are considered: two-parameter semigroups or processes and the skew product flows. Their attractors are defined in terms of families of sets that are mapped onto each other under the dynamics rather than a single set as in autonomous systems. Two types of attraction are now possible: pullback attraction, which depends on the behaviour from the system in the distant past, and forward attraction, which depends on the behaviour of the system in the distant future. These are generally independent of each other.The component subsets of pullback and forward attractors exist in actual time. The asymptotic behaviour in the future limit is characterised by omega-limit sets, in terms of which form what are called forward attracting sets. They are generally not invariant in the conventional sense, but are asymptotically invariant in general and, if the future dynamics is appropriately uniform, also asymptotically negatively invariant.Much of this book is based on lectures given by the authors in Frankfurt and Wuhan. It was written mainly when the first author held a 'Thousand Expert' Professorship at the Huazhong University of Science and Technology in Wuhan.

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Page Count

157

Publisher

World Scientific

Published Date

ISBN 10

9811228671

ISBN 13

9789811228674

Taylor Approximations for Stochastic Partial Differential Equations

By: Arnulf Jentzen,Peter E. Kloeden

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Taylor Approximations for Stochastic Partial Differential Equations

By: Arnulf Jentzen,Peter E. Kloeden

This book presents a systematic theory of Taylor expansions of evolutionary-type stochastic partial differential equations (SPDEs). The authors show how Taylor expansions can be used to derive higher order numerical methods for SPDEs, with a focus on pathwise and strong convergence. In the case of multiplicative noise, the driving noise process is assumed to be a cylindrical Wiener process, while in the case of additive noise the SPDE is assumed to be driven by an arbitrary stochastic process with Hl̲der continuous sample paths. Recent developments on numerical methods for random and stochastic ordinary differential equations are also included since these are relevant for solving spatially discretised SPDEs as well as of interest in their own right. The authors include the proof of an existence and uniqueness theorem under general assumptions on the coefficients as well as regularity estimates in an appendix.

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Page Count

234

Publisher

SIAM

Published Date

ISBN 10

1611972019

ISBN 13

9781611972016

Stochastic Climate Models

By: Peter Imkeller,Jin-Song Von Storch

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Stochastic Climate Models

By: Peter Imkeller,Jin-Song Von Storch

A collection of articles written by mathematicians and physicists, designed to describe the state of the art in climate models with stochastic input. Mathematicians will benefit from a survey of simple models, while physicists will encounter mathematically relevant techniques at work.

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Page Count

446

Publisher

Springer Science & Business Media

Published Date

ISBN 10

376436520X

ISBN 13

9783764365202

Book's by Peter E. Kloeden

Numerical Solution of Stochastic Differential Equations

Numerical Solution of Stochastic Differential Equations

Chaotic Numerics

Chaotic Numerics

Recent Developments in Computational Finance

Recent Developments in Computational Finance

Metric Spaces of Fuzzy Sets

Metric Spaces of Fuzzy Sets

An Introduction to the Numerical Simulation of Stochastic Diﬀerential Equations

An Introduction to the Numerical Simulation of Stochastic Diﬀerential Equations

Nonautonomous Dynamical Systems

Nonautonomous Dynamical Systems

Random Ordinary Differential Equations and Their Numerical Solution

Random Ordinary Differential Equations and Their Numerical Solution

An Introduction To Nonautonomous Dynamical Systems And Their Attractors

An Introduction To Nonautonomous Dynamical Systems And Their Attractors

Taylor Approximations for Stochastic Partial Differential Equations

Taylor Approximations for Stochastic Partial Differential Equations

Stochastic Climate Models

Stochastic Climate Models

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