Full history recursive multilevel Picard approximations for ordinary differential equations with expectations

03/03/2021 ∙ by Christian Beck, et al. ∙ 0

We consider ordinary differential equations (ODEs) which involve expectations of a random variable. These ODEs are special cases of McKean-Vlasov stochastic differential equations (SDEs). A plain vanilla Monte Carlo approximation method for such ODEs requires a computational cost of order ε^-3 to achieve a root-mean-square error of size ε. In this work we adapt recently introduced full history recursive multilevel Picard (MLP) algorithms to reduce this computational complexity. Our main result shows for every δ>0 that the proposed MLP approximation algorithm requires only a computational effort of order ε^-(2+δ) to achieve a root-mean-square error of size ε.

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