Institut Élie Cartan de Lorraine

About: Institut Élie Cartan de Lorraine is a facility organization based out in Vandœuvre-lès-Nancy, France. It is known for research contribution in the topics: Boundary value problem & Stochastic differential equation. The organization has 345 authors who have published 1084 publications receiving 15512 citations. The organization is also known as: Institut Élie-Cartan de Nancy.

Simulation of hitting times for Bessel processes with non-integer dimension

Abstract: In this paper, we complete and improve the study of the simulation of the hitting times of some given boundaries for Bessel processes. These problems are of great interest in many application fields as finance and neurosciences. In a previous work (Ann. Appl. Probab. 23 (2013) 2259–2289), the authors introduced a new method for the simulation of hitting times for Bessel processes with integer dimension. The method, called walk on moving spheres algorithm (WoMS), was based mainly on the explicit formula for the distribution of the hitting time and on the connection between the Bessel process and the Euclidean norm of the Brownian motion. This method does not apply anymore for a non-integer dimension. In this paper we consider the simulation of the hitting time of Bessel processes with non-integer dimension $\delta\geq1$ and provide a new algorithm by using the additivity property of the laws of squared Bessel processes. We split each simulation step of the algorithm in two parts: one is using the integer dimension case and the other one considers hitting time of a Bessel process starting from zero.

Etude et analyse numérique des transferts de chaleur couplés par rayonnement et conduction dans les milieux semi-transparents : application aux milieux fibreux

Abstract: L'objet de ce travail est l'etude et l'analyse numerique des transferts de chaleur couples par rayonnement et conduction a travers les milieux semi-transparents. Le modele utilise est constitue d'un systeme de deux equations aux derivees partielles couplees : l'equation integro-differentielle du transfert radiatif (ETR), qui a comme inconnue la luminance, et une equation non lineaire de la chaleur regissant la temperature dans le milieu. Dans le premier chapitre de la these, nous detaillons la modelisation avec les hypotheses simplificatrices qu'elle comporte. Dans le second chapitre, nous montrons l'existence et l'unicite du systeme couple d'equations en regime stationnaire. Le troisieme chapitre est consacre a la resolution numerique des equations en regime stationnaire. Pour resoudre l'ETR, nous discretisons l'espace angulaire suivant plusieurs directions et nous utilisons une quadrature numerique pour approcher l'integrale de l'equation. Il en resulte alors un systeme differentiel lineaire du premier ordre que nous resolvons par trois methodes differentes. La deuxieme equation est resolue a l'aide d'un schema aux differences finies, associe a une transformation de Kirchhoff. Le couplage entre les deux equations est resolu par une methode de point fixe. Dans le quatrieme chapitre, nous etudions la convergence d'un schema numerique en regime stationnaire. Dans le cinquieme chapitre nous presentons une methode numerique pour resoudre le systeme couple en regime transitoire, d'une part lorsque les temperatures sont imposees aux frontieres et, d'autre part, lorsque le milieu est soumis a des conditions de flux. L'equation de la chaleur est resolue en espace par la methode des elements finis P2. Le systeme differentiel en temps est resolu par une methode de Runge-Kutta implicite, adaptee aux equations raides. Le dernier chapitre de ce travail analyse les resultats numeriques obtenus par la simulation appliquee a un materiau isolant constitue de fibres de silice.

Topological sensitivity analysis for elliptic problems on graphs

Abstract: We consider elliptic problems on graphs under given loads and bilateral contact conditions. We ask the question which graph is best suited to sustain the loads and the constraints. More precisely, given a cost function we may look at a multiple node of the graph with edge degree $N$ and ask as to whether that node should be resolved into a number of node of edge degree 3, in order to decrease the cost. Thus, we are looking into the sensitivity of a graph carrying an elliptic problem with respect to changing the topology of the graph by releasing nodes with high edge degree. In order words, we are looking into the topological gradient of an elliptic problem on a graph.

Rate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noise

Abstract: We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\in(1/3,1)$ and multiplicative noise component $\sigma$. When $\sigma$ is constant and for every $H\in(0,1)$, it was proved in [Ann. Probab. 33 (2005) 703–758] that, under some mean-reverting assumptions, such a process converges to its equilibrium at a rate of order $t^{-\alpha}$ where $\alpha\in(0,1)$ (depending on $H$). In [Ann. Inst. Henri Poincare Probab. Stat. 53 (2017) 503–538], this result has been extended to the multiplicative case when $H>1/2$. In this paper, we obtain these types of results in the rough setting $H\in(1/3,1/2)$. Once again, we retrieve the rate orders of the additive setting. Our methods also extend the multiplicative results of [Ann. Inst. Henri Poincare Probab. Stat. 53 (2017) 503–538] by deleting the gradient assumption on the noise coefficient $\sigma$. The main theorems include some existence and uniqueness results for the invariant distribution.

An alternative expression for the Black-Scholes formula in terms of Brownian first and last passage times

Abstract: The celebrated Black-Scholes formula which gives the price of a European option, may be expressed as the cumulative function of a last passage time of Brownian motion. A related result involving first passage times is also obtained.