MC 6460
Speaker
Peter KloedenÌý | Universitat Tubingen, Germany
Title
Random ordinary differential equations and their numerical approximation
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Random ordinary differential equations (RODEs)Ìý are pathwise ordinary differential equations that containÌý a stochastic process inÌý their vector field functions. TheyÌýÌý have been used for many years in a wide range of applications,Ìý but have been very much overshadowed byÌý stochastic ordinary differential equations (SODEs). TheÌý stochastic process could be a fractional Brownian motion orÌý a Poisson process, but when it is a diffusion process then there is aÌý close connection between RODEs and SODEs through the Doss-Sussmann transformation and its generalisations, whichÌý relate aÌý RODE and an SODE with the same (transformed) solutions. RODEsÌý play an important role in the theory of random dynamical systems and random attractors.ÌýÌý
Classical numerical schemes such as Runge-Kutta schemesÌý can be used forÌý RODEs but do not achieve their usual high order since the vector fieldÌý does not inherit enough smoothnessÌý in time from the driving process. It will be shown how, nevertheless,Ìý various kinds of Taylor-likeÌý expansions of the solutions of RODESÌý can be obtained when the stochastic process has Hölder continuous or even measurable sample paths and then used to deriveÌý pathwiseÌý convergent numerical schemes of arbitrarily high order. The use of bounded noise and anÌý application in biology will be considered.Ìý