Download or read book Numerical Approximations of Stochastic Differential Equations with Non Globally Lipschitz Continuous Coefficients written by Martin Hutzenthaler and published by . This book was released on 2015 with total page 99 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method diverge for these SDEs in finite time. This article develops a general theory based on rare events for studying integrability properties such as moment bounds for discrete-time stochastic processes. Using this approach, we establish moment bounds for fully and partially drift-implicit Euler methods and for a class of new explicit approximation methods which require only a few more arithmetical operations than the Euler-Maruyama method. These moment bounds are then used to prove strong convergence of the proposed schemes. Finally, we illustrate our results for several SDEs from finance, physics, biology and chemistry.

Download or read book Numerical Approximations of Stochastic Differential Equations with Non Globally Lipschitz Continuous Coefficients written by Martin Hutzenthaler and published by American Mathematical Soc.. This book was released on 2015-06-26 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method diverge for these SDEs in finite time. This article develops a general theory based on rare events for studying integrability properties such as moment bounds for discrete-time stochastic processes. Using this approach, the authors establish moment bounds for fully and partially drift-implicit Euler methods and for a class of new explicit approximation methods which require only a few more arithmetical operations than the Euler-Maruyama method. These moment bounds are then used to prove strong convergence of the proposed schemes. Finally, the authors illustrate their results for several SDEs from finance, physics, biology and chemistry.

Download or read book Taylor Approximations for Stochastic Partial Differential Equations written by Arnulf Jentzen and published by SIAM. This book was released on 2011-12-08 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 H?lder 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.

Download or read book Numerical Methods for Stochastic Partial Differential Equations with White Noise written by Zhongqiang Zhang and published by Springer. This book was released on 2017-09-01 with total page 391 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book covers numerical methods for stochastic partial differential equations with white noise using the framework of Wong-Zakai approximation. The book begins with some motivational and background material in the introductory chapters and is divided into three parts. Part I covers numerical stochastic ordinary differential equations. Here the authors start with numerical methods for SDEs with delay using the Wong-Zakai approximation and finite difference in time. Part II covers temporal white noise. Here the authors consider SPDEs as PDEs driven by white noise, where discretization of white noise (Brownian motion) leads to PDEs with smooth noise, which can then be treated by numerical methods for PDEs. In this part, recursive algorithms based on Wiener chaos expansion and stochastic collocation methods are presented for linear stochastic advection-diffusion-reaction equations. In addition, stochastic Euler equations are exploited as an application of stochastic collocation methods, where a numerical comparison with other integration methods in random space is made. Part III covers spatial white noise. Here the authors discuss numerical methods for nonlinear elliptic equations as well as other equations with additive noise. Numerical methods for SPDEs with multiplicative noise are also discussed using the Wiener chaos expansion method. In addition, some SPDEs driven by non-Gaussian white noise are discussed and some model reduction methods (based on Wick-Malliavin calculus) are presented for generalized polynomial chaos expansion methods. Powerful techniques are provided for solving stochastic partial differential equations. This book can be considered as self-contained. Necessary background knowledge is presented in the appendices. Basic knowledge of probability theory and stochastic calculus is presented in Appendix A. In Appendix B some semi-analytical methods for SPDEs are presented. In Appendix C an introduction to Gauss quadrature is provided. In Appendix D, all the conclusions which are needed for proofs are presented, and in Appendix E a method to compute the convergence rate empirically is included. In addition, the authors provide a thorough review of the topics, both theoretical and computational exercises in the book with practical discussion of the effectiveness of the methods. Supporting Matlab files are made available to help illustrate some of the concepts further. Bibliographic notes are included at the end of each chapter. This book serves as a reference for graduate students and researchers in the mathematical sciences who would like to understand state-of-the-art numerical methods for stochastic partial differential equations with white noise.

Download or read book Numerical Solution of Stochastic Differential Equations written by Peter E. Kloeden and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 666 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Download or read book Multidimensional Diffusion Processes written by Daniel W. Stroock and published by Springer. This book was released on 2007-02-03 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the reviews: "This book is an excellent presentation of the application of martingale theory to the theory of Markov processes, especially multidimensional diffusions. [...] This monograph can be recommended to graduate students and research workers but also to all interested in Markov processes from a more theoretical point of view." Mathematische Operationsforschung und Statistik

Download or read book Monte Carlo and Quasi Monte Carlo Methods written by Art B. Owen and published by Springer. This book was released on 2018-07-03 with total page 476 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the refereed proceedings of the Twelfth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing that was held at Stanford University (California) in August 2016. These biennial conferences are major events for Monte Carlo and quasi-Monte Carlo researchers. The proceedings include articles based on invited lectures as well as carefully selected contributed papers on all theoretical aspects and applications of Monte Carlo and quasi-Monte Carlo methods. Offering information on the latest developments in these very active areas, this book is an excellent reference resource for theoreticians and practitioners interested in solving high-dimensional computational problems, arising in particular, in finance, statistics, computer graphics and the solution of PDEs.

Download or read book Exact Finite Difference Schemes written by Sergey Lemeshevsky and published by Walter de Gruyter GmbH & Co KG. This book was released on 2016-09-26 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt: Exact Finite-Difference Schemes is a first overview of the topic also describing the state-of-the-art in this field of numerical analysis. Construction of exact difference schemes for various parabolic and elliptic partial differential equations are discussed, including vibrations and transport problems. After this, applications are discussed, such as the discretisation of ODEs and PDEs and numerical methods for stochastic differential equations. Contents: Basic notation Preliminary results Hyperbolic equations Parabolic equations Use of exact difference schemes to construct NSFD discretizations of differential equations Exact and truncated difference schemes for boundary-value problem Exact difference schemes for stochastic differential equations Numerical blow-up time Bibliography

Download or read book Random Ordinary Differential Equations and Their Numerical Solution written by Xiaoying Han and published by Springer. This book was released on 2017-10-25 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Monte Carlo and Quasi Monte Carlo Methods written by Aicke Hinrichs and published by Springer Nature. This book was released on with total page 657 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book An Introduction to Stochastic Differential Equations with Reflection written by Andrey Pilipenko and published by Universitätsverlag Potsdam. This book was released on 2014 with total page 90 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Numerical Solutions of Stochastic Differential Equations written by Liguo Wang and published by . This book was released on 2016 with total page 141 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this dissertation, we consider the problem of simulation of stochastic differential equations driven by Brownian motions or the general Lévy processes. There are two types of convergence for a numerical solution of a stochastic differential equation, the strong convergence and the weak convergence. We first introduce the strong convergence of the tamed Euler-Maruyama scheme under non-globally Lipschitz conditions, which allow the polynomial growth for the drift and diffusion coefficients. Then we prove a new weak convergence theorem given that the drift and diffusion coefficients of the stochastic differential equation are only twice continuously differentiable with bounded derivatives up to order 2 and the test function are third order continuously differentiable with all of its derivatives up to order 3 satisfying a polynomial growth condition. We also introduce the multilevel Monte Carlo method, which is efficient in reducing the total computational complexity of computing the expectation of a functional of the solution of a stochastic differential equation. This method combines the three sides of the simulation of stochastic differential equations: the strong convergence, the weak convergence and the Monte Carlo method. At last, a recent progress of the strong convergence of the numerical solutions of stochastic differential equations driven by Lévy processes under non-globally Lipschitz conditions is also presented.

Download or read book Stochastic Numerics for Mathematical Physics written by Grigori N. Milstein and published by Springer Nature. This book was released on 2021-12-03 with total page 754 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a substantially revised and expanded edition reflecting major developments in stochastic numerics since the first edition was published in 2004. The new topics, in particular, include mean-square and weak approximations in the case of nonglobally Lipschitz coefficients of Stochastic Differential Equations (SDEs) including the concept of rejecting trajectories; conditional probabilistic representations and their application to practical variance reduction using regression methods; multi-level Monte Carlo method; computing ergodic limits and additional classes of geometric integrators used in molecular dynamics; numerical methods for FBSDEs; approximation of parabolic SPDEs and nonlinear filtering problem based on the method of characteristics. SDEs have many applications in the natural sciences and in finance. Besides, the employment of probabilistic representations together with the Monte Carlo technique allows us to reduce the solution of multi-dimensional problems for partial differential equations to the integration of stochastic equations. This approach leads to powerful computational mathematics that is presented in the treatise. Many special schemes for SDEs are presented. In the second part of the book numerical methods for solving complicated problems for partial differential equations occurring in practical applications, both linear and nonlinear, are constructed. All the methods are presented with proofs and hence founded on rigorous reasoning, thus giving the book textbook potential. An overwhelming majority of the methods are accompanied by the corresponding numerical algorithms which are ready for implementation in practice. The book addresses researchers and graduate students in numerical analysis, applied probability, physics, chemistry, and engineering as well as mathematical biology and financial mathematics.

Download or read book Local Lipschitz Continuity in the Initial Value and Strong Completeness for Nonlinear Stochastic Differential Equations written by Sonja Cox and published by American Mathematical Society. This book was released on 2024-05-15 with total page 102 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Download or read book Recent Developments in Computational Finance written by Thomas Gerstner and published by World Scientific. This book was released on 2013 with total page 481 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Numerical Approximations of Stochastic Maxwell Equations written by Chuchu Chen and published by Springer Nature. This book was released on 2024-01-04 with total page 293 pages. Available in PDF, EPUB and Kindle. Book excerpt: The stochastic Maxwell equations play an essential role in many fields, including fluctuational electrodynamics, statistical radiophysics, integrated circuits, and stochastic inverse problems. This book provides some recent advances in the investigation of numerical approximations of the stochastic Maxwell equations via structure-preserving algorithms. It presents an accessible overview of the construction and analysis of structure-preserving algorithms with an emphasis on the preservation of geometric structures, physical properties, and asymptotic behaviors of the stochastic Maxwell equations. A friendly introduction to the simulation of the stochastic Maxwell equations with some structure-preserving algorithms is provided using MATLAB for the reader’s convenience. The objects considered in this book are related to several fascinating mathematical fields: numerical analysis, stochastic analysis, (multi-)symplectic geometry, large deviations principle, ergodic theory, partial differential equation, probability theory, etc. This book will appeal to researchers who are interested in these topics.

Download or read book Invariant Measures for Stochastic Nonlinear Schr dinger Equations written by Jialin Hong and published by Springer Nature. This book was released on 2019-08-22 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides some recent advance in the study of stochastic nonlinear Schrödinger equations and their numerical approximations, including the well-posedness, ergodicity, symplecticity and multi-symplecticity. It gives an accessible overview of the existence and uniqueness of invariant measures for stochastic differential equations, introduces geometric structures including symplecticity and (conformal) multi-symplecticity for nonlinear Schrödinger equations and their numerical approximations, and studies the properties and convergence errors of numerical methods for stochastic nonlinear Schrödinger equations. This book will appeal to researchers who are interested in numerical analysis, stochastic analysis, ergodic theory, partial differential equation theory, etc.