Mean-square convergence of numerical methods for random ordinary differential equations

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The aim of this work is to analyze the mean-square convergence rates of numerical schemes for random ordinary differential equations (RODEs). First, a relation between the global and local mean-square convergence order of one-step explicit approximations is established. Then, the global mean-square convergence rates are investigated for RODE-Taylor schemes for general RODEs, Affine-RODE-Taylor schemes for RODEs with affine noise, and Ito-Taylor schemes for RODEs with Ito noise, respectively. The theoretical convergence results are demonstrated through numerical experiments.