Applied Math Seminar | Peter Kloeden, Random ordinary differential equations and their numerical approximation

MC 6460

Speaker

Peter Kloeden  | Universitat Tubingen, Germany

Title

Random ordinary differential equations and their numerical approximation

 Abstract

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 or  a Poisson process, but when it is a diffusion process then 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.  

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,  various kinds of Taylor-like  expansions of the solutions of RODES  can be obtained when the stochastic process has Hölder continuous or even measurable sample paths and then used to derive  pathwise  convergent numerical schemes of arbitrarily high order. The use of bounded noise and an  application in biology will be considered.