Contact Info
Department of Applied Mathematics
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
Phone: 519-888-4567, ext. 32700
Fax: 519-746-4319
PDF files require Adobe Acrobat Reader
MC 6460
Peter Kloeden | Universitat Tubingen, Germany
Random ordinary differential equations and their numerical approximation
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.
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Contact Info
Department of Applied Mathematics
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
Phone: 519-888-4567, ext. 32700
Fax: 519-746-4319
PDF files require Adobe Acrobat Reader
The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is co-ordinated within the Office of Indigenous Relations.