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We show that the probability of the exceptional set decays exponentially fora broad class of randomized algorithms approximating solutions of ODEs,admitting a certain error decomposition. This class includes randomizedexplicit and implicit Euler schemes, and the randomized two-stage Runge-Kuttascheme (under inexact information). We design a confidence interval for theexact solution of an IVP and perform numerical experiments to illustrate thetheoretical results.

On the properties of the exceptional set for the randomized Euler and Runge-Kutta schemes