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

Abstract

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 ${\epsilon }^{-3}$ to achieve a root-mean-square error of size $\epsilon$. 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 $\delta >0$ that the proposed MLP approximation algorithm requires only a computational effort of order ${\epsilon }^{-(2+\delta )}$ to achieve a root-mean-square error of size $\epsilon$.

arXiv comment24 pages. arXiv admin note: substantial text overlap with arXiv:1903.05985

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Cited by in corpus (1)

• Multilevel Picard approximations for McKean-Vlasov stochastic differential equations