Contenu / Programme
UE Mathématiques Appliquées
Stochastic calculus and applications: This course is devoted to the introduction of the basic concepts of continuous time stochastic processes which are used in many fields : physics, finance, biology, medicine, filtering theory, decision theory. It will consist of a presentation of Brownian motion, Itô integral, stochastic differential equations and Girsanov theorem. Several applications will be given.
– Financial markets
– Brownian motion
– Continuous time martingales
– Stochastic calculus
– Black-Sholes model
– Interest rates
C. Bernardin : Professor of Probability UCA R. Catellier : Associate Professor of Probability UCA
UE Mathématiques pour la Modélisation
Probabilistic numerical methods: Probabilistic numerical methods are widely used in machine learning algorithms as well as in mathematical finance for pricing financial derivatives and computing strategies. The course will present the basic methods used for simulating random variables and implementing the Monte-Carlo methods. Simulation in Scilab of stochastic processes used in mathematical finance, such as Brownian motion and solutions to stochastic differential equations, will be discussed as well.
– Random variables simulations
– Non-asymptotic estimate of approximation error
– Central Limit Theorem
– Berry-Esseen Theorem
– Variance reduction technique
– Stochastic differential equations discretisation
– Euler and Milstein scheme
– Romberg-Richardson extrapolation
– Stochastic algorithms
E. Tanré : Researcher in Probability INRIA Sophia
PROGRAMME MSS 2021-2022 7
Advanced statistics: This course focuses on three pillars of modern statistical inference: parameter estimation, hypothesis testing, and model selection. Its aim is to provide a good understanding of the current methods via a thorough treatment of the existing theoretical guarantees. A particular emphasis will be placed on the asymptotic setting.
– Introduction
probability theory, a quick reminder
stochastic convergences, usual probability distributions (esp. exponential family)
– Statistical estimation
– M- and Z-estimators: consistency, asymptotic normality
– local average estimators (nearest neighbor rule, histograms). – consistency
– information inequalities (Cramer-Rao, Fisher)
– asymptotic efficiency
– introduction to U-statistics
– Statistical testing
– reminders (usual tests, type I and II error, p-value)
– multiple testing (Bonferroni correction, Benjamini-Hochberg) – relative efficiency, asymptotic efficiency (Bahadur)
– safe testing (Gruenwald)
– Model selection
– introduction: Mallows Cp heuristic
– penalised least-squares and oracle inequalities – the Lasso (recent results)
D. Garreau : Associate Professor of Statistics UCA
Modeling Studies: In this course the student will apply the theoretical knowledge obtained during the courses of stochastic calculus, probabilistic computational methods and advanced statistics in some applied contexts.
Part 1: The goal of this part is to understand a technical report or a short conference paper. Using the knowledge from Probability and Statistics, the student will learn how to model a real-life data by point processes. The completed list of papers will be given at least one week before the beginning of the course. References: Daley, D.J, Vere-Jones, D. An introduction to the theory of Point Processes, Volume 1.
Part 2: Volatility modeling . The objective of this part is to introduce the students in the modeling beyond the Black Scholes model. To build models that replicate features of the stock market that are being observed and price (exotic) derivatives by applying numerical methods.
PROGRAMME MSS 2021-2022 8
Syllabus of Part 2:
– Local (deterministic) volatility. Backing out local volatility surface from European option prices, Dupire model.
– Stochastic volatility, Fourier transform and Heston model.
– Pricing exotic options (forward start option, Asian, look back, etc).
– The Black Scholes model and its limitation,
