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# Loughborough University Institutional Repository

 Please use this identifier to cite or link to this item: https://dspace.lboro.ac.uk/2134/36716

 Title: Numerical analysis of random periodicity of stochastic differential equations Authors: Liu, Yu Keywords: Random periodic solutionPeriodic measureEuler–Maruyama methodModified Milstein methodInfinite horizonRate of convergencePull-backWeak convergence Issue Date: 2018 Publisher: © Yu Liu Abstract: In this thesis, we discuss the numerical approximation of random periodic solutions (r.p.s.) of stochastic differential equations (SDEs) with multiplicative noise. We prove the existence of the random periodic solution as the limit of the pull-back flow when the starting time tends to $-\infty$ along the multiple integrals of the period. As the random periodic solution is not explicitly constructible, it is useful to study the numerical approximation. We discretise the SDE using the Euler-Maruyama scheme and modified Milstein scheme. Subsequently we obtain the existence of the random periodic solution as the limit of the pull-back of the discretised SDE. We prove that the latter is an approximated random periodic solution with an error to the exact one at the rate of $\sqrt {\Delta t}$ in the mean-square sense in Euler-Maruyama method and $\Delta t$ in the modified Milstein method. We obtain the weak convergence result in infinite horizon for the approximation of the average periodic measure. Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University. Sponsor: Loughborough University (Development Fund). URI: https://dspace.lboro.ac.uk/2134/36716 Appears in Collections: PhD Theses (Maths)

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