Ce cours a pour objectif d’introduire les principaux outils de base en programmation dynamique en restant dans un cadre déterministe et en insistant sur les applications économiques. Nous traiterons dans un premier temps du temps discret en horizon fini puis infini. Nous aborderons ensuite le temps continu avec le calcul des variations et une introduction au contrôle optimal en insistant sur l’approche programmation dynamique de Bellman. Certaines preuves seront juste ébauchées de manière heuristique sans que soit explicité en détail le cadre fonctionnel rigoureux.
Definition and properties of multi-dimensional normal distribution, definition and elementary properties of Brownian motion, existence and Hölder continuity of Brownian motion using the Kolmogorov-Chentsov continuity criterion, filtrations, stopping times, progressive measurability, path properties, martingales, uniform integrability, Vitali's convergence theorem, sub- and supermartingales, maximum inequality, Doob's inequality for p-integrable submartingales, Doob's optional sampling theorem with applications, local martingales and examples, integration of predictable step processes, p-variation of functions, quadratic variation and covariation process of continuous local martingales, Kunita-Watanabe inequality, stochastic integration for continuous local martingales and generalization for continuous semimartingales, chain rule and convergence theorems for stochastic integrals (with respect to continuous semimartingales), integration by parts, multi-dimensional Ito formula with applications, Tanaka's formula, local Ito formula and Ito formula for holomorphic functions.
Slides: [PDF]
Stochastic optimal control: basic results and applications [PDF]
Stochastic optimal control: optimal stopping [PDF]
Stochastic optimal control: relaxed formulation [PDF]
Stochastic optimal control: viscosity solutions [PDF], [PDF]
Stochastic optimal control: irreversible investment model [PDF]
Stochastic optimal control: superreplication cost in uncertain volatility model [PDF]
BSDE: reflected BSDE [PDF]
BSDE: weak formulation of control problem [PDF]
BSDE: maximum principle [PDF]
BSDE application: heading and weak formulation of control [PDF]
BSDE application: exponential utility maximization in complete market [PDF]
BSDE application: portfolio optimization under constraints [PDF]
BSDE application: Mean-variance criterion for portfolio selection [PDF]
Monte-Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results of the problems which are difficult or impossible to solve analytically. We focus on the following topics in this course.
Introduction [PDF]
Generating random variables [PDF], [Generate Gaussian RV, Notebook], [Acceptance-Rejection, Notebook], [Copula, Notebook]
Convergence, error estimates (law of large number, central limit theorem, concentration inequalities) [PDF], [General MC Notebook]
Variance reduction [PDF], [Variance reduction, Notebook]
Some basic result about Brownian motion, Itô's integral w.r.t. Brownian motion, Itô's formula
Basic existence, uniqueness, stability results of stochastic differential equations and properties of the solutions
Feynman-Kac theorems, the relation between SDEs and PDEs [PDF]
Numerics of SDEs: Euler-Maruyama scheme, Milstein scheme, high-order schemes, Multi-level Monte-Carlo [PDF]
Statistical error in the simulation of SDEs [PDF]
Simulation of non-linear processes, backward SDEs and non-linear versions of Feynman-Kac theorems
McKean-Vlasov SDE and simulation
Stochastic optimization
Markov chain Monte-Carlo
[Exercise + Solution], [Python Antithetic sampling Notebook], [Python Antithetic sampling PDF], [Python Importance sampling Notebook], [Python Importance sampling PDF], [Python Black-Scholes Notebook], [Python Black-Scholes PDF]
[Exercise + Solution], [Python BM construction Notebook], [Python BM construction PDF]
[Exercise + Solution], [Python Numerics SDE Notebook], [Python Numerics SDE PDF]
After successful completion of the course, students are able to explain how Machine Learning works and how it is applied to mathematical problems in finance. In particular, the students are able to
decide and justify whether machine learning is suitable for solving a given problem in finance,
formalize the problem and select a suitable algorithm for the solution,
evaluate the result.
Abstract neurons [PDF]
Forward neural network [PDF]
Activation functions [PDF]
Cost functions [PDF]
Training errors and overfitting [PDF]
Optimization [PDF]
Universal approximation [PDF]
Backpropagation algorithm [PDF], [Toy example PDF], [Toy example Notebook]
Wavelet frames [PDF]
Deep neural networks with wavelet [PDF]
Universal interpolation and controlled ODEs [PDF]
Weights initialization [PDF]
Some python examples [MNIST Notebook], [MNIST Html]
Deep heading [PDF], [PDF], [Binomial PDF], [Binomial Notebook], [Black-Scholes PDF], [Black-Scholes Notebook]
Deep portfolio optimization [Portfolio PDF], [Portfolio Notebook]
Ein-Periodenmodell (Arbitrage, risikoneutrales Maß, Bewertung von Claims, Vollständigkeit, Sub-/Superhedging, optimale Portfolios) Mehrperiodenmodell in diskreter Zeit (selbstfinanzierende Handelsstrategien, Satz von Dalang/Morton/Willinger) Binomialmodell, Verteilung des Maximums, Grenzübergang im Binomialmodell, Black-Scholes Modell, Amerikanische Optionen, Snell'sche Einhüllende, Doob'sche Zerlegung
Die Vorlesung basiert auf dem Buch „Stochastic Finance: an Introduction in Discrete Time“ von Föllmer und Schied.
https://www.degruyter.com/document/doi/10.1515/9783110218053/html