The V AMMCS International Conference

Waterloo, Ontario, Canada | August 18-23, 2019

AMMCS 2019 Plenary Talk

Random ordinary differential equations and their numerical approximation

P.E. Kloeden (University of Tuebingen)

Random ordinary differential equations (RODEs) are pathwise ordinary differential equations that contain a stochastic process in their vector field functions. They have been used for many years in a wide range of applications, but have been very much overshadowed by stochastic ordinary differential equations (SODEs). The stochastic process could be a fractional Brownian motion, but when it is a diffusion process there is a close connection between RODEs and SODEs through the Doss-Sussmann transformation and its generalisations, which relate a RODE and an SODE with the same (transformed) solutions. RODEs play an important role in the theory of random dynamical systems and random attractors. They are also useful in biology.

Classical numerical schemes such as Runge-Kutta schemes can be used for RODEs but do not achieve their usual high order since the vector field does not inherit enough smoothness in time from the driving process. It will be shown how, nevertheless, Taylor expansions of the solutions of RODES can be obtained when the stochastic process has Hoelder continuous sample paths and then used to derive pathwise  convergent numerical schemes of arbitrarily high order. RODEs with Ito noise will also be considered as well as RODEs with afine structure and Poisson noise. Applications to biology in will be given.

Xiaoying Han and P. E. Kloeden, Random Ordinary Differential Equations and their Numerical Solution,  Springer Nature Singapore, 2017.
Peter Kloeden completed his Ph.D. and D.Sc. at the University of Queensland, Australia in 1975 and 1995.  He was until recently a professor of mathematics at the Goethe University in Frankfurt am Main and then research professor of mathematics at the Huazhong University of Science & Technology in China. He now lives in Tuebingen. He has wide interests in the applications of mathematical analysis, numerical analysis, stochastic analysis and dynamical systems.  Professor Kloeden 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)   and "Random Ordinary Differential Equationsand Their Numerical Solution” (with Xiaoying Han) published by Springer in 1992 and 2017.  He is a Fellow of the Society of Industrial and Applied Mathematics and was awarded the W.T. & Idalia Reid Prize in 2006.  His current interests focus on nonautonomous and random dynamical systems and their applications in the biological sciences.