An Introduction To The Numerical Simulation Of Stochastic Differential Equations
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An Introduction to the Numerical Simulation of Stochastic Di?erential Equations
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.
Stochastic Numerical Methods
Stochastic Numerical Methods introduces at Master level the numerical methods that use probability or stochastic concepts to analyze random processes. The book aims at being rather general and is addressed at students of natural sciences (Physics, Chemistry, Mathematics, Biology, etc.) and Engineering, but also social sciences (Economy, Sociology, etc.) where some of the techniques have been used recently to numerically simulate different agent-based models. Examples included in the book range from phase-transitions and critical phenomena, including details of data analysis (extraction of critical exponents, finite-size effects, etc.), to population dynamics, interfacial growth, chemical reactions, etc. Program listings are integrated in the discussion of numerical algorithms to facilitate their understanding. From the contents: Review of Probability Concepts Monte Carlo Integration Generation of Uniform and Non-uniform Random Numbers: Non-correlated Values Dynamical Methods Applications to Statistical Mechanics Introduction to Stochastic Processes Numerical Simulation of Ordinary and Partial Stochastic Differential Equations Introduction to Master Equations Numerical Simulations of Master Equations Hybrid Monte Carlo Generation of n-Dimensional Correlated Gaussian Variables Collective Algorithms for Spin Systems Histogram Extrapolation Multicanonical Simulations
Numerical Solution of Stochastic Differential Equations
Author: Peter E. Kloeden
language: en
Publisher: Springer Science & Business Media
Release Date: 2013-04-17
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.