"Deep Artificial Neural Network Approximations for High-dimensional PDEs and Optimal Stopping Problems"Welti, TimoIn recent years deep artificial neural networks (DNNs) have very successfully been employed in numerical simulations for a multitude of computational problems including, for example, object and face recognition, natural language processing, fraud detection, computational advertisement, and numerical approximations of partial differential equations (PDEs). Such numerical simulations indicate that DNNs seem to be able to overcome the curse of dimensionality in the sense that the number of real parameters used to describe the DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy and the dimension of the function which the DNN aims to approximate in such computational problems.While there is a large number of rigorous mathematical approximation results for artificial neural networks in the scientific literature, there are only a few special situations where results in the literature can rigorously explain the success of DNNs when approximating high-dimensional functions. In the first part of this talk it is revealed that DNNs do indeed overcome the curse of dimensionality in the numerical approximation of Kolmogorov PDEs with constant diffusion and nonlinear drift coefficients. A crucial ingredient in our proof of this result is the fact that the artificial neural network used to approximate the solution of the PDE really is a deep artificial neural network with a large number of hidden layers. The second part of this talk concerns a new deep learning based method for solving high-dimensional optimal stopping problems, which generally also suffer from the curse of dimensionality. Among several possible application scenarios, the introduced algorithm can, in particular, be employed for the pricing of American options with a large number of underlyings, where it allows to approximatively compute both the price as well as an optimal exercise strategy. Numerical results on benchmark problems, which suggest that the proposed algorithm is highly effective in the case of many underlyings in terms of both accuracy and speed, conclude the talk. This talk is based on joint works with Sebastian Becker, Patrick Cheridito, Arnulf Jentzen, and Diyora Salimova. |
| https://arxiv.org/abs/1809.07321 |
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