Random Ordinary Differential Equations and Their Numerical Solution

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.

• 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.

“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)