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Author
Date
2020Type
- Doctoral Thesis
ETH Bibliography
yes
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Abstract
In virtually any scenario where there is a desire to make quantitative or qualitative predictions, mathematical models are of crucial importance for predicting quantities of interest. Unfortunately, many models are based on mathematical objects that cannot be calculated exactly. As a consequence, numerical approximation algorithms that approximate such objects are indispensable in practice. These algorithms are often stochastic in nature, which means that there is some form of randomness involved when running them. In this thesis we consider four possibly high-dimensional approximation problems and corresponding stochastic numerical approximation algorithms. More specifically, we prove essentially sharp rates of convergence in the probabilistically weak sense for spatial spectral Galerkin approximations of semi-linear stochastic wave equations driven by multiplicative noise. In addition, we develop an abstract framework that allows us to view full-history recursive multilevel Picard approximation methods from a new perspective and to work out more clearly how these stochastic methods beat the curse of dimensionality in the numerical approximation of semi-linear heat equations. Furthermore, we tackle high-dimensional optimal stopping problems by proposing a stochastic numerical approximation algorithm that is based on deep learning. And finally, we study convergence in the probabilistically strong sense of the overall error arising in deep learning based empirical risk minimisation, one of the main pillars of supervised learning. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000457220Publication status
publishedExternal links
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Contributors
Examiner: Jentzen, Arnulf
Examiner: Grohs, Philipp
Examiner: Kloeden, Peter
Examiner: Mishra, Siddhartha
Publisher
ETH ZurichSubject
weak convergence; stochastic wave equations; multiplicative noise; full-history recursive multilevel Picard approximations; curse of dimensionality; semi-linear partial differential equations; PDEs; semi-linear heat equations; American option; Bermudan option; financial derivative; derivative pricing; option pricing; optimal stopping; deep learning; deep neural networks; empirical risk minimisation; full error analysis; approximation; generalisation; optimisation; strong convergence; stochastic gradient descent; random initialisationOrganisational unit
02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics
Funding
ETH-47 15-2 - Mild stochastic calculus and numerical approximations for nonlinear stochastic evolution equations with Levy noise (ETHZ)
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ETH Bibliography
yes
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