Calcul stochastique

Le calcul est l’étude des phénomènes aléatoires dépendant du temps. À ce titre, c'est une extension de la théorie des probabilités. Ne pas confondre avec la technique des calculateurs stochastiques. Applications Le domaine d’application du calcul stochastique comprend la mécanique quantique, le traitement du signal, la chimie, les mathématiques financières, la météorologie et même la musique. Processus aléatoires Un processus aléatoire X est une famille de variables aléatoires indexée par un sous-ensemble de \R ou \N, souvent assimilé au temps (voir aussi Processus stochastique). C'est une fonction de deux variables : le temps et l'état de l'univers \omega. L'ensemble des états de l'univers est traditionnellement noté \Omega. L'application qui à un \omega fixé associe X(\omega,t), t variable, est appelée trajectoire du processus ; c'est une simple fonction

The non-linear stochastic wave equation in high dimensions

Daniel Conus

The main topic of this thesis is the study of the non-linear stochastic wave equation in spatial dimension greater than 3 driven by spatially homogeneous Gaussian noise that is white in time. We are interested in questions of existence and uniqueness of solutions, as well as in properties of solutions, such as existence of high order moments and Hölder-continuity properties. The stochastic wave equation is formulated as an integral equation in which appear stochastic integrals with respect to martingale measures (in the sense of J.B. Walsh). Since, in dimensions greater than 3, the fundamental solution of the wave equation is neither a function nor a non-negative measure, but a general Schwartz distribution, we first develop an extension of the Dalang-Walsh stochastic integral that makes it possible to integrate a wide class of Schwartz distributions. This class contains the fundamental solution of the wave equation, under a hypothesis on the spectral measure of the noise that has already been used in the literature. With this extended stochastic integral, we establish existence of a square-integrable random-field solution to the non-linear stochastic wave equation in any dimension. Uniqueness of the solution is established within a specific class of processes. In the case of a fine multiplicative noise, we obtain a series representation of the solution and estimates on the p-th moments of the solution (p ≥ 1). From this, we deduce Hölder-continuity of the solution under standard assumptions. The Hölder exponent that we obtain is optimal. For the case of general multiplicative noise, we construct a framework for working with appropriate iterated stochastic integrals and then derive a truncated Itô-Taylor expansion for the solution of the stochastic wave equation. The convergence of this expansion remains an open problem, so we conclude with some remarks that suggest an Itô-Taylor series expansion for the solution.

EPFL2008

The non-linear stochastic wave equation in high dimensions

Daniel Conus

EPFL2008

Lecture Notes on Stochastic Calculus (Part II)

The non-linear stochastic wave equation in high dimensions

Lecture Notes on Stochastic Calculus (Part I)

The non-linear stochastic wave equation in high dimensions

Lecture Notes on Stochastic Calculus (Part II)

Lecture Notes on Stochastic Calculus (Part I)