Mathematical Modelling

LIST OF LECTURERS
François Delarue (UNS, local coordinator)
LIST OF LECTURES (6 ECTS each)
Stochastic Calculus and mathematical finance. Prof. Cédric Bernardin (UNS), 1st Trimester
This course is devoted to the introduction of the basic concepts used in mathematical finance. It will consist of a presentation of Brownian motion, Itô integral, stochastic differential equations and Girsanov theorem. From the modelling point of view, all these tools will be used to introduce the notions of strategy, arbitrage and riskneutral probability measure and to define the Black Scholes model used for the pricing of European options. A good knowledge in probability theory (with measure theory) is required.
Prerequisities: Probability theory with measure theory.
Statistical inference in the regression setting. Prof. Yannick Baraud (UNS), 1st Trimester
The regression framework is a powerful modelling tool for understanding how a quantitative variable (the stock price,the income of a firm, the sales of a product…) varies as time goes by or depends on a set predictors. The aim of this course is to present some mathematical tools for statistical inference in this setting. We shall deal with the problems of nonparametric estimation, variable selection and testing the goodness of fit of a model, among others. Our point of view will mainly be nonasymptotic and based on a model selection approach. For illustration, we shall provide an application to finance and the problem of estimating the drift in the BlackScholes model where this parameter is allowed to depend on time.
Prerequisities: Probability theory with measure theory.
Numerical methods in probability for mathematical finance. Prof. Etienne Tanré and James Inglis (INRIA), 1st and 2nd Trimesters
Probabilistic numerical methods are widely used 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 MonteCarlo method. Simulation of stochastic processes used in mathematical finance, such as Brownian motion and solutions to stochastic differential equations, will be discussed as well.
Prerequisities: Probability theory with measure theory, Stochastic Calculus
Stochastic control in mathematical finance. Prof. François Delarue (UNS), 2nd Trimester
This course will be an introduction to the theory of optimal stochastic control, which is widely used in mathematical finance for portfolio optimization and option pricing. Lectures will consist of a description of mathematical tools for characterizing the optimal strategies: dynamic programming principle and HamiltonJacobiBellman equations, stochastic Pontryagin principle and backward stochastic differential equations.
Prerequisities: Probability theory with measure theory, Stochastic Calculus
Parabolic methods in finance. Numerical methods. Prof. Roland Masson (UNS), 2nd Trimester
This course will focus on the use of parabolic PDEs in mathematical finance. A first part of the lectures will consist of a reformulation of the Black Scholes model and of the option pricing problem in terms of PDE methods. Then, advanced numerical methods, based on finite volume methods, will be discussed.
Prerequisities: Differential calculus in finite dimension, Basic elements of functional analysis.
MathMods programme Université de Nice – Sophia Antipolis (France)