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We consider ordinary differential equations (ODEs) which involve expectationsof a random variable. These ODEs are special cases of McKean-Vlasov stochasticdifferential equations (SDEs). A plain vanilla Monte Carlo approximation methodfor such ODEs requires a computational cost of order $\varepsilon^{-3}$ toachieve a root-mean-square error of size $\varepsilon$. In this work we adaptrecently introduced full history recursive multilevel Picard (MLP) algorithmsto reduce this computational complexity. Our main result shows for every$\delta>0$ that the proposed MLP approximation algorithm requires only acomputational effort of order $\varepsilon^{-(2+\delta)}$ to achieve aroot-mean-square error of size $\varepsilon$.

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