Bergische Universität Wuppertal Fachbereich Mathematik und Naturwissenschaften Angewandte Mathematik - Numerische Analysis (AMNA)

People Research Publications Teaching

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The lecture is suitable for students of mathematics as well as for economics.
The students of economathematics can use it as component AKap.NAaA-a "Selected Topics in Numerical Analysis and Algorithms" in the module of the same name.

Topics of the Lecture:
The aim of the lecture is to introduce the mathematical concepts underlying the theory of approximating a function using machine learning, resp. to be more precise deep learning methods. To this end we give an introduction to the theory and show how this can be used for exploring Neural Networks. To illustrate the applicability of Neural Networks in Finance we consider the pricing of derivatives and the calibration of parametric models. For pricing we consider the Black-Scholes-Merton, the Hull-White and the Heston model. Calibration is considered for the Black-Scholes-Merton and the Heston model.

Outline of the lecture:

1. Introduction to Learning Theory
   • Error Decomposition
   • PAC (probably approximately correct) Bounds
   • Growth Functions, VC (Vapnik-Chervonenkis) dimension, Rademacher Complexity and Extensions
   • Algorithmic Stability and further PAC Bounds
2. Pricing Financial Derivatives
   • Short Introduction to Option Pricing
   • Black-Scholes-Merton, Hull-White and Heston models
3. Neural Networks
   • Definition of MLP (Multi Layer Perceptrons)
   • Approximation Theory
   • VC dimension and Rademacher Complexity of Neural Networks
   • Training of Neural Networks
   • (Stochastic) Gradient Descent
   • Backpropagation
   • Applications to Finance (Pricing, Calibration)

Literature:

• S. Bubeck and Nicolo Cesa-Bianchi, Regret Analysis of Stochastic and Nonstochastic Multi-armed Bandit Problems, Now Publishers Incorporate, 2012.
• N. Cesa-Bianchi and G. Lugosi, Prediction, Learning, and Games, Cambridge University Press, 2006.
• C. Giraud, Introduction to High-Dimensional Statistics Chapman and Hall / CRC, 2014.
• T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer, 2nd edition, 2009.
• G. James, D. Witten, T. Hastie and R. Tibshirani, An Introduction to Statistical Learning, Springer.
• V. Koltchinskii, Oracle Inequalities in Empirical Risk Minimization and Sparse Recovery Problems, Springer, 2011.
• T. Mitchell, Machine Learning, McGraw-Hill, 1997.
• M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning (2nd edition), MIT Press.
• S. Shalev-Shwartz and S. Ben-David, Understanding Machine Learning: From Theory to Algorithms, Cambridge University Press, 2014.
• P. Wilmottt, Machine Learning: An Applied Mathematics Introduction, 2019.

Previous knowledge: Analysis I - III, basic knowledge of ordinary differential equations, stochastics.

Exercises:
For the exercises we recommend to install the Python distribution from Anaconda and install the packages tensorflow, keras, pytorch, matplotlib. If other packages are necessary these can easily installed on demand.
Sheets.

Criteria: Regular participation and participation in the exercise groups, as well as reaching 50% of the possible points on the first seven or the remaining exercise sheets and at least 2/3 of the possible points for the practical tasks.