# Philippe von WurstembergerETH Zurich | ETH Zürich · Department of Mathematics

Philippe von Wurstemberger

## About

11

Publications

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Introduction

**Skills and Expertise**

## Publications

Publications (11)

In financial engineering, prices of financial products are computed approximately many times each trading day with (slightly) different parameters in each calculation. In many financial models such prices can be approximated by means of Monte Carlo (MC) simulations. To obtain a good approximation the MC sample size usually needs to be considerably...

In this paper we develop a new machinery to study the capacity of artificial neural networks (ANNs) to approximate high-dimensional functions without suffering from the curse of dimensionality. Specifically, we introduce a concept which we refer to as approximation spaces of artificial neural networks and we present several tools to handle those sp...

One of the most challenging issues in applied mathematics is to develop and analyze algorithms which are able to approximately compute solutions of high-dimensional nonlinear partial differential equations (PDEs). In particular, it is very hard to develop approximation algorithms which do not suffer under the curse of dimensionality in the sense th...

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 determine fair prices of financial derivatives in the financial industry. The PDEs appearing in financial engineering applications are often nonlinear...

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...

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...

For a long time it is 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 dimensionali...

For a long time it is 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 dimen-sional...

The stochastic gradient descent (SGD) optimization algorithm plays a central role in a series of machine learning applications. The scientific literature provides a vast amount of upper error bounds for the SGD method. Much less attention as been paid to proving lower error bounds for the SGD method. It is the key contribution of this paper to make...

Stochastic gradient descent (SGD) optimization algorithms are key ingredients in a series of machine learning applications. In this article we perform a rigorous strong error analysis for SGD optimization algorithms. In particular, we prove for every arbitrarily small ε ∈ (0, ∞) and every arbitrarily large p ∈ (0, ∞) that the considered SGD optimiz...

## Projects

Project (1)

Partial differential equations (PDEs) are among the most universal tools used in modeling problems in nature and man-made complex systems. In particular, PDEs are a fundamental tool in portfolio optimization problems and in the state-of-the-art pricing and hedging of financial derivatives. The PDEs appearing in such financial engineering applications are often high dimensional as the dimensionality of the PDE corresponds to the number of financial asserts in the involved hedging portfolio. Such PDEs can typically not be solved explicitly and developing efficient numerical algorithms for high dimensional PDEs is one of the most challenging tasks in applied mathematics. As is well-known, the difficulty lies in the so-called ``curse of dimensionality'' in the sense that the computational effort of standard approximation algorithms grows exponentially in the dimension of the considered PDE and there is only a very limited number of cases where a practical PDE approximation algorithm with a computational effort which grows at most polynomially in the PDE dimension has been developed. It is the key objective of this research project to overcome this curse of dimensionality and to construct and analyze new approximation algorithms which solve high dimensional PDEs with a computational effort that grows at most polynomially in both the dimension of the PDE and the reciprocal of the prescribed approximation precision. Key tools, which we intend to use to achieve this objective, are suitable nonlinear Feynman-Kac formulas and deep neural networks.