The book is interdisciplinary in its appoach and orientation
It places equal emphasis on both theory and applications
Besides serving as a basic text on stochastic differential equations it derives and discusses the numerical methods needed to solve such equations
The book is interdisciplinary in its appoach and orientation
It places equal emphasis on both theory and applications
Besides serving as a basic text on stochastic differential equations it derives and discusses the numerical methods needed to solve such equations
Includes supplementary material: sn.pub/extras
The numerical analysis of stochastic differential equations (SDEs) differs significantly from that of ordinary differential equations, due to the peculiarities of stochastic calculus. This book provides an easily accessible introduction to SDEs, their applications and the numerical methods to solve such equations. To help the reader develop an intuitive understanding and hands-on numerical skills, numerous exercises and PC-Exercises are included. The book is directed at a multi-disciplinary readership, consisting primarily of engineers, financial analysts, physicists and mathematicians developing numerical schemes for applications of SDEs, and also of researchers in other fields like biology, chemistry or economics who, with less mathematical background, wish to apply
1. Probability and Statistics.- 2. Probability and Stochastic Processes.- 3. Ito Stochastic Calculus.- 4. Stochastic Differential Equations.- 5. Stochastic Taylor Expansions.- 6. Modelling with Stochastic Differential Equations.- 7. Applications of Stochastic Differential Equations.- 8. Time Discrete Approximation of Deterministic Differential Equations.- 9. Introduction to Stochastic Time Discrete Approximation.- 10. Strong Taylor Approximations.- 11. Explicit Strong Approximations.- 12. Implicit Strong Approximations.- 13. Selected Applications of Strong Approximations.- 14. Weak Taylor Approximations.- 15. Explicit and Implicit Weak Approximations.- 16. Variance Reduction Methods.- 17. Selected Applications of Weak Approximations.- Solutions of Exercises.- Bibliographical Notes.
The numerical analysis of stochastic differential equations differs significantly from that of ordinary differential equations due to peculiarities of stochastic calculus. This book provides an introduction to stochastic calculus and stochastic differential equations, in both theory and applications, emphasising the numerical methods needed to solve such equations. It assumes of the reader an undergraduate background in mathematical methods typical of engineers and physicists, though many chapters begin with a deillegalscriptive summary. The book is also accessible to others who only require numerical recipes. The stochastic Taylor expansion provides the basis for the discrete time numerical methods for differential equations. The book presents many new results on high-order methods for strong sample path approximations and for weak functional approximations, including implicit, predictor-corrector, extra-polation and variance-reduction methods. Besides serving as a basic text on such methods, the book offers the reader ready access to a large number of potential research problems in a field that is just beginning to expand rapidly and is widely applicable. To help the reader to develop an intuitive understanding of the underlying mathematics and hand-on numerical skills, exercises and over 100 PC-Exercises are included.
"... 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. This was not an easy task... Their exposition stresses clarity, not formality - a very welcome approach." ZAMP
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.
The book presents many new results on higher-order methods for strong sample path approximations and for weak functional approximations, including implicit, predictor-corrector, extrapolation and variance-reduction methods. Besides serving as a basic text on such methods, the book offers the reader ready access to a large number of potential resarch problems in a field that is just beginning to expand rapidly and is widely applicable.
Numerische Analysis stochastischer Differentialgleichungen unterscheidet sich deutlich von gewöhnlichen Differentialgleichungen durch die Besonderheiten stochastischen Rechnens. Das vorliegende Buch bietet Lesern mit "undergraduate" Kenntnissen eine leicht zugängliche Einführung in Stochastischen Differentialgleichungen, Anwendungen und Numerische Methoden für Physik und Technik. Zahlreiche Übungen sollen intuitives Verstehen und numerisches Geschick des Lesers fördern. Es wendet sich an Ingenieure, Physiker und Mathematiker die numerische Schemata für Anwendungen von Stochastischen Differentialgleichungen entwickeln und an Wissenschaftler aus anderen Gebieten, wie Biologie, Chemie oder Wirtschaftswissenschaften, mit weniger mathematischen Kenntnissen, die bestehende numerische Methoden für ihre eigene Arbeit verwenden wollen.
Professor Eckhard Platen is a joint appointment between the School of Finance and Economics and the Department of Mathematical Sciences to the 1997 created Chair in Quantitative Finance at the University of Technology Sydney. Prior to this appointment he was Founding Head of the Centre for Financial Mathematics at the Institute of Advanced Studies at the Australian National University in Canberra. He completed a PhD in Mathematics at the Technical University in Dresden in 1975 and obtained in 1985 his Dr. sc. from the Academy of Sciences in Berlin, where he headed at the Weierstrass Institute the Sector of Stochastics. He is co-author of two successful books on Numerical Methods for Stochastic Differential Equations, published by Springer Verlag, and has authored more than 100 research papers in quantitative finance and mathematics.
Inhaltsverzeichnis
1. Probability and Statistics.- 2. Probability and Stochastic Processes.- 3. Ito Stochastic Calculus.- 4. Stochastic Differential Equations.- 5. Stochastic Taylor Expansions.- 6. Modelling with Stochastic Differential Equations.- 7. Applications of Stochastic Differential Equations.- 8. Time Discrete Approximation of Deterministic Differential Equations.- 9. Introduction to Stochastic Time Discrete Approximation.- 10. Strong Taylor Approximations.- 11. Explicit Strong Approximations.- 12. Implicit Strong Approximations.- 13. Selected Applications of Strong Approximations.- 14. Weak Taylor Approximations.- 15. Explicit and Implicit Weak Approximations.- 16. Variance Reduction Methods.- 17. Selected Applications of Weak Approximations.- Solutions of Exercises.- Bibliographical Notes.
Klappentext
The aim of this book is to provide an accessible introduction to stochastic differ ential equations and their applications together with a systematic presentation of methods available for their numerical solution. During the past decade there has been an accelerating interest in the de velopment of numerical methods for stochastic differential equations (SDEs). This activity has been as strong in the engineering and physical sciences as it has in mathematics, resulting inevitably in some duplication of effort due to an unfamiliarity with the developments in other disciplines. Much of the reported work has been motivated by the need to solve particular types of problems, for which, even more so than in the deterministic context, specific methods are required. The treatment has often been heuristic and ad hoc in character. Nevertheless, there are underlying principles present in many of the papers, an understanding of which will enable one to develop or apply appropriate numerical schemes for particular problems or classes of problems.
The numerical analysis of stochastic differential equations (SDEs) differs significantly from that of ordinary differential equations, due to the peculiarities of stochastic calculus. This book provides an easily accessible introduction to SDEs, their applications and the numerical methods to solve such equations. To help the reader develop an intuitive understanding and hands-on numerical skills, numerous exercises and PC-Exercises are included. The book is directed at a multi-disciplinary readership, consisting primarily of engineers, financial analysts, physicists and mathematicians developing numerical schemes for applications of SDEs, and also of researchers in other fields like biology, chemistry or economics who, with less mathematical background, wish to apply
