Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational advertising to numerical approximations of partial differential equations (PDEs). Such numerical simulations suggest that ANNs have the capacity to

For a long time it has been well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semilinear heat equations with gradient-independent and globally Lipschitz continuous nonlinearities that the computational effort of a variant of the recently introduced multilevel Picard approximations grows at most polynomially both in the dimension and in the reciprocal of the required accuracy.

Parabolic partial differential equations (PDEs) are widely used in the mathematical modeling of natural phenomena and man-made complex systems. In particular, parabolic PDEs are a fundamental tool to approximately determine fair prices of financial derivatives in the financial engineering industry. The PDEs appearing in financial engineering applications are often nonlinear (e.g., in PDE models which take into account the possibility of a defaulting counterparty) and high-dimensional since the dimension typically corresponds to the number of considered financial assets. A major issue in the scientific literature is that most approximation methods for nonlinear PDEs suffer from the so-called curse of dimensionality in the sense that the computational effort to compute an approximation with a prescribed accuracy grows exponentially in the dimension of the PDE or in the reciprocal of the prescribed approximation accuracy and nearly all approximation methods for nonlinear PDEs in the scientific literature have not been shown not to suffer from the curse of dimensionality. Recently, a new class of approximation schemes for semilinear parabolic PDEs, termed full history recursive multilevel Picard (MLP) algorithms, were introduced and it was proven that MLP algorithms do overcome the curse of dimensionality for semilinear heat equations. In this paper we extend and generalize those findings to a more general class of semilinear PDEs which includes as special cases the important examples of semilinear Black-Scholes equations used in pricing models for financial derivatives with default risks. In particular, we introduce an MLP algorithm for the approximation of solutions of semilinear Black-Scholes equations and prove, under the assumption that the nonlinearity in the PDE is globally Lipschitz continuous, that the computational effort of the proposed method grows at most polynomially in both the

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.