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Abstract: We consider the weak convergence of numerical methods for stochasticdifferential equations SDEs. Weak convergence is usually expressed in termsof the convergence of expected values of test functions of the trajectories.Here we present an alternative formulation of weak convergence in terms of thewell-known Prokhorov metric on spaces of random variables. For a general classof methods, we establish bounds on the rates of convergence in terms of theProkhorov metric. In doing so, we revisit the original proofs of weakconvergence and show explicitly how the bounds on the error depend on thesmoothness of the test functions. As an application of our result, we use theStrassen - Dudley theorem to show that the numerical approximation and the truesolution to the system of SDEs can be re-embedded in a probability space insuch a way that the method converges there in a strong sense. One corollary ofthis last result is that the method converges in the Wasserstein distance,another metric on spaces of random variables. Another corollary establishesrates of convergence for expected values of test functions assuming only localLipschitz continuity. We conclude with a review of the existing results forpathwise convergence of weakly converging methods and the corresponding strongresults available under re-embedding.



Autor: Benoit Charbonneau, Yuriy Svyrydov, P.F. Tupper

Fuente: https://arxiv.org/



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