Random Ordinary Differential Equations and Their Numerical Solution: (85 Probability Theory and Stochastic Modelling)

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. 

Table of Contents:
Preface.- Reading Guide.- Part I Random and Stochastic Ordinary Differential Equations.- 1.Introduction.-. 2.Random ordinary differential equations.- 3.Stochastic differential equations.- 4.Random dynamical systems.- 5.Numerical dynamics.- Part II Taylor Expansions.- 6.Taylor expansions for ODEs and SODEs.- 7.Taylor expansions for RODEs with affine noise.- 8.Taylor expansions for general RODEs.- Part III Numerical Schemes for Random Ordinary Differential Equations.- 9.Numerical methods for ODEs and SODEs.- 10.Numerical schemes: RODEs with Itô noise.- 11.Numerical schemes: affine noise.- 12.RODE–Taylor schemes.- 13.Numerical stability.- 14.Stochastic integrals.- Part IV Random Ordinary Differential Equations in the Life Sciences.- 15.Simulations of biological systems.- 16.Chemostat.- 17.Immune system virus model.- 18.Random Markov chains.- Part V Appendices.- A.Probability spaces.- B.Chain rule for affine RODEs.- C.Fractional Brownian motion.- References.- Index.

About the Author :
Professor Peter E. Kloeden has wide interests in the applications of mathematical analysis, numerical analysis, stochastic analysis and dynamical systems.  He is the coauthor of several influential books on nonautonomous dynamical systems, metric spaces of fuzzy sets, and in particular “Numerical Solutions of Stochastic Differential equations” (with E. Platen) published by Springer in 1992.  Professor Kloeden is a Fellow of the Australian Mathematical Society and the Society of Industrial and Applied Mathematics. He was awarded the W.T. & Idalia Reid Prize from Society of Applied and Industrial Mathematics in 2006.  His current interests focus on nonautonomous and random dynamical systems and their applications in the biological sciences. Professor Xiaoying Han’s main research interests are in random and nonautonomous dynamical systems and their applications.  In addition to mathematical analysis of dynamical systems, she is also interested in modeling and simulation of applied dynamical systems in biology, chemical engineering, ecology, material sciences, etc.  She is the coauthor of the books “Applied Nonautonomous and Random Dynamical Systems” (with T. Caraballo) and “Attractors under Discretisation” (with P. E. Kloeden), published in the SpringerBrief series.

Review :
“This monograph provides a general overview of random ordinary differential equations (RODEs) with emphasis on numerical methods used to solve them. … This book is a welcome addition to the literature that will be especially helpful to scholars seeking an understanding of RODEs and the numerical methods used to solve them.” (Melvin D. Lax, zbMATH 1392.60003, 2018)