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Full-history recursive multilevel Picard (MLP) approximation schemes havebeen shown to overcome the curse of dimensionality in the numericalapproximation of high-dimensional semilinear partial differential equations(PDEs) with general time horizons and Lipschitz continuous nonlinearities.However, each of the error analyses for MLP approximation schemes in theexisting literature studies the $L^2$-root-mean-square distance between theexact solution of the PDE under consideration and the considered MLPapproximation and none of the error analyses in the existing literatureprovides an upper bound for the more general $L^p$-distance between the exactsolution of the PDE under consideration and the considered MLP approximation.It is the key contribution of this article to extend the $L^2$-error analysisfor MLP approximation schemes in the literature to a more general $L^p$-erroranalysis with $p\in (0,\infty)$. In particular, the main result of this articleproves that the proposed MLP approximation scheme indeed overcomes the curse ofdimensionality in the numerical approximation of high-dimensional semilinearPDEs with the approximation error measured in the $L^p$-sense with $p \in(0,\infty)$.

Strong $L^p$-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations