Author: "Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS)" - Searchworks@Jio Institute Digital Library Search Results

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63 results on '"Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS)"'

1. Statistical and renewal results for the random sequential adsorption model applied to a unidirectional multicracking problem

Author: Pierre Vallois, Pierre Calka, André Mézin, Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de science et génie des surfaces ( LSGS ), Institut National Polytechnique de Lorraine ( INPL ) -Centre National de la Recherche Scientifique ( CNRS ), Institut Élie Cartan de Nancy ( IECN ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Université Henri Poincaré - Nancy 1 ( UHP ) -Université Nancy 2-Institut National Polytechnique de Lorraine ( INPL ) -Centre National de la Recherche Scientifique ( CNRS ), Probabilistic numerical methods ( OMEGA ), Inria Sophia Antipolis - Méditerranée ( CRISAM ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Université Henri Poincaré - Nancy 1 ( UHP ) -Université Nancy 2-Centre National de la Recherche Scientifique ( CNRS ), Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de science et génie des surfaces (LSGS), Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)
Subjects: Statistics and Probability, Composite material, Stationary process, Relaxation of stress, Markov chain, Stationary processes, 01 natural sciences, Point process, Packing model, Coating, Fibre, 010104 statistics & probability, Modelling and Simulation, Poisson point process, Renewal theory, Random sequential adsorption, 0101 mathematics, Real line, Mathematics, Rupture, Crack, Stochastic process, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Palm measure, Conditional probability distribution, Renewal process, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Distribution (mathematics), Car-parking model, Modeling and Simulation, ComputingMethodologies_GENERAL, Brittleness, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR]
Abstract: We work out a stationary process on the real line to represent the positions of the multiple cracks which are observed in some composites materials submitted to a fixed unidirectional stress ɛ . Our model is the one-dimensional random sequential adsorption. We calculate the intensity of the process and the distribution of the inter-crack distance in the Palm sense. Moreover, the successive crack positions of the one-sided process (denoted by X i ɛ , i ⩾ 1 ) are described. We prove that the sequence { ( X i ɛ , Y i ɛ ) , 1 ⩽ i ⩽ n } is a “conditional renewal process”, where Y i ɛ is the value of the stress at which X i ɛ forms. The approaches “in the Palm sense” and “one-sided process” merge when n → + ∞ . The saturation case ( ɛ =+ ∞ ) is also investigated.
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2. Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time, II

Author: Bernard Roynette, Marc Yor, Pierre Vallois, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Probabilités et Modèles Aléatoires (LPMA), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
Subjects: [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Combinatorics, 010104 statistics & probability, General Mathematics, 010102 general mathematics, Limiting, 0101 mathematics, 01 natural sciences, Brownian motion, Mathematics
Abstract: Let us denote by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $P_0$ \end{document} the Wiener measure defined on the canonical space \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\big(\Omega=\mathcal{C}(\mathbb{R}_+, \mathbb{R})$ \end{document}, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} ${(X_t)}_{t \ge 0}$ \end{document}, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} ${(\mathcal{F}_t)}_{t \ge 0}\big)$ \end{document}, and let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(S_t)$ \end{document} (resp. \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(I_t)$ \end{document}), be the one sided-maximum (resp. minimum), \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(L^0_t)$ \end{document} the local time at \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $0$ \end{document}, and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(D_t)$ \end{document} the number of down-crossings from \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $b$ \end{document} to \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $a$ \end{document} (with \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $b≯a$ \end{document}). Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $f\: \mathbb{R} \times \mathbb{R}^d \to \left]0, +\infty\right[$ \end{document} be a Borel function, and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(A_t)$ \end{document} be a process chosen within the set \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\big\{(S_t)$ \end{document}; \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(S_t,t)$ \end{document}; \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(L^0_t)$ \end{document}; \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(S_t,I_t,L^0_t)$ \end{document}; \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(D_t)\big\}$ \end{document}, which consists of 5 elements. We prove a penalization result: under suitable assumptions on \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $f$ \end{document}, there exists a positive \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\big((\mathcal{F}_t),P_0\big)$ \end{document}-martingale \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(M_t^{f})$ \end{document}, starting at \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $1$ \end{document}, such that \begin{equation} \label{ab1} \lim_{t\to \infty}\frac{E_0 \big[1_{\Gamma_s}f(X_t,A_t)\big]}{E_0\big[f(X_t,A_t)\big]}=Q_0^{f}(\Gamma_s):= E_0 [1_{\Gamma_s} M_s^{f}],\quad \forall \Gamma_s \in \mathcal{F}_s\ \text{and}\ s\ge 0. \end{equation} We determine the law of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(X_t)$ \end{document} under the p.m. \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $Q_0^{f}$ \end{document} defined on \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(\Omega, \mathcal{F}_\infty)$ \end{document} by \eqref{ab1}. For the \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $1^{\rm st}, 3^{\rm rd}$ \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $5^{\rm th}$ \end{document} elements of the set, we prove first that \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $Q_0^{f} (A_{\infty} < \infty)=1$ \end{document}, and more generally \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $Q_0^{f} (0 where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \(g= \sup\, \{s≯0$ \end{document}, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $A_s=A_{\infty}\}$ \end{document} \vspace{2pt} (with the convention \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sup \emptyset=0$ \end{document}). Secondly, we split the trajectory of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(X_t)$ \end{document} into \vspace{-2pt} two parts: \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} ${(X_t)}_{0\le t \le g}$ \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} ${(X_{t+g})}_{ t \ge 0}$ \end{document}, and we describe their laws under \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $Q_0^{f}$ \end{document}, conditionally on \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $A_{\infty}$ \end{document}. For the \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $2^{\rm nd}$ \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $4^{\rm th}$ \end{document} \vspace{2pt} elements, a similar result holds replacing \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $A_\infty$ \end{document} by resp. \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $S_\infty$ \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $S_\infty \vee I_\infty$ \end{document}.
Published: 2006

3. Range of Brownian Motion with Drift

Author: Etienne Tanré, Pierre Vallois, Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)
Subjects: [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Statistics and Probability, 010104 statistics & probability, Range (mathematics), Mathematics::Probability, General Mathematics, 010102 general mathematics, Mathematical analysis, 0101 mathematics, Statistics, Probability and Uncertainty, 01 natural sciences, Brownian motion, Mathematics
Abstract: Let (B (t); t 0) be a Brownian motion with drift > 0, starting at 0. Let us define by induction S1 = inf t 0 B (t), 1 the last time such that
Published: 2006

4. m-order integrals and generalized Itô's formula; the case of a fractional Brownian motion with any Hurst index

Author: Mihai Gradinaru, Ivan Nourdin, Pierre Vallois, Francesco Russo, Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS)
Subjects: Statistics and Probability, Fractional Brownian motion, m-order integral, 010102 general mathematics, Mathematical analysis, Type (model theory), 60H05, 60G15, 60G18, Heavy traffic approximation, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Mathematics::Probability, Real-valued function, Probability theory, Integer, Itô's formula, Bounded function, Mathematics [G03] [Physical, chemical, mathematical & earth Sciences], Mathématiques [G03] [Physique, chimie, mathématiques & sciences de la terre], 0101 mathematics, Statistics, Probability and Uncertainty, Mathematics, Probability measure, Mathematical physics
Abstract: Given an integer m, a probability measure ν on [0,1], a process X and a real function g, we define the m-order ν-integral having as integrator X and as integrand g(X). In the case of the fractional Brownian motion B, for any locally bounded function g, the corresponding integral vanishes for all odd indices m>1/2H and any symmetric ν. One consequence is an Itô–Stratonovich type expansion for the fractional Brownian motion with arbitrary Hurst index 01/6.
Published: 2005

5. On long time behavior of some coagulation processes

Author: Bernard Roynette, Etienne Tanré, Nicolas Fournier, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS)
Subjects: Statistics and Probability, Particle system, Coalescence (physics), Stochastic process, Differential equation, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Coalescence, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Modelling and Simulation, Modeling and Simulation, Almost surely, 0101 mathematics, Non-linear stochastic differential equations with jumps, Jump process, Brownian motion, Excitation, Mathematics
Abstract: We consider an infinite system of particles characterized by their position and mass, in which coalescence occurs. Each particle endures Brownian excitation, and is subjected to the attraction of a potential. We define a stochastic process ( X t , M t ) t ⩾0 describing the evolution of the position and mass of a typical particle. We show that under some conditions, the mass process M t tends almost surely to infinity, while the position process X t tends almost surely to 0, as time tends to infinity.
Published: 2004

6. Parameter estimation of pharmacokinetics models in the presence of timing noise

Author: Muriel Barberi-Heyob, Nacim Ramdani, Pierre Vallois, Sophie Mézières-Wantz, Thierry Bastogne, Centre de Recherche en Automatique de Nancy (CRAN), Université Henri Poincaré - Nancy 1 (UHP)-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria), Centre d'Etudes et Recherches en Thermique, Environnement et Systèmes [Créteil] (CERTES EA 3481), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), and Vallois, Pierre
Subjects: [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], 0303 health sciences, Computer science, Estimation theory, health care facilities, manpower, and services, education, Uptake kinetics, Variance (accounting), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 03 medical and health sciences, Noise, 0302 clinical medicine, Control theory, 030220 oncology & carcinogenesis, health care economics and organizations, 030304 developmental biology
Abstract: The problem addressed in this paper deals with the parameter estimation of in vitro uptake kinetics of drugs into living cells in presence of timing noise. Effects of the timing noise on the bias and variance of the output error are explicitly determined. A bounded-error parameter estimation approach is proposed as a suited solution to handle this problem. Application results are presented and emphasize its effectiveness in such an experimental framework.
Published: 2007

7. Quelques approximations du temps local brownien

Author: Pierre Vallois, Blandine Berard Bergery, Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)
Subjects: 60G44, 60H05, 60H99, 60J55, 60J65, Pure mathematics, Approximations of π, temps local, 010102 general mathematics, Probability (math.PR), vitesse de convergence, Geometry, General Medicine, 01 natural sciences, AMS 60G44, 60H05, 60H99, 60J55, 60J65, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, intégration stochastique par régularisation, Mathematics::Probability, Local time, variation quadratique, FOS: Mathematics, 0101 mathematics, Brownian motion, Mathematics - Probability, Mathematics
Abstract: We give some approximations of the local time process $(L_t^x)_{t\geqslant 0}$ at level $x$ of the real Brownian motion $(X_t)$. We prove that $ \frac{2}{\epsilon}\int_0^{t} X_{(u+\epsilon)\wedge t}^+ \indi_{\{X_u \leqslant 0\}} du + \frac{2}{\epsilon}\int_0^{t} X_{(u+\epsilon) \wedge t}^- \indi_{\{X_u>0\}} du$ and $\frac{4}{\epsilon}\int_0^{t} X_u^- \indi_{\{X_{(u+\epsilon) \wedge t} > 0\}} du$ converge in the ucp sense to $L_t^0$, as $\epsilon \to 0$. We show that $ \frac{1}{\epsilon}\int_0^t (\indi_{\{x, Comment: Soumis dans les Comptes rendus - Math\'ematique
Published: 2007

8. Stochastic Particle Method Applied to Local Wind Simulation

Author: Antoine Rousseau, T. Salameh, Frédéric Bernardin, Mireille Bossy, Philippe Drobinski, Modelling, Observations, Identification for Environmental Sciences (MOISE), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Météorologie Dynamique (UMR 8539) (LMD), Département des Géosciences - ENS Paris, École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École des Ponts ParisTech (ENPC)-École polytechnique (X)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC), IEEE, Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-École polytechnique (X)-École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Département des Géosciences - ENS Paris, École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École normale supérieure - Paris (ENS-PSL), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)
Subjects: Physics, [PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], Mathematical optimization, Wind power, 010504 meteorology & atmospheric sciences, business.industry, Stochastic process, Differential equation, 020209 energy, Numerical analysis, 02 engineering and technology, Computational fluid dynamics, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Stochastic differential equation, Lagrangian stochastic model, 0202 electrical engineering, electronic engineering, information engineering, Applied mathematics, MM5, business, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], 0105 earth and related environmental sciences
Abstract: International audience; A Lagrangian stochastic model is introduced in order to describe the local behavior of the wind. Based on some MM5 data, and thanks to particles driven by stochastic differential equations, we propose a numerical method allowing to improve the MM5 simulations at small scales, without requiring too much additional computational cost.
Published: 2007

9. An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient function of the form |x|^a, a in [1/2,1)

Author: Bossy, Mireille, Diop, Awa, Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), and INRIA
Subjects: [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], weak error, CIR process, Hull-White process, Euler scheme, SABR processes, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: Version 2; In this paper, we study the rate of convergence of a symmetrized version of the classical Euler scheme, applied to the discretisation of the solution of a stochastic differential equation with a diffusion coefficient function of the form |x|^a, a in [1/2,1). For smooth test functions, we show that the weak error is of order one as for the classical Euler scheme. More over, the symmetrized version is very easy to simulate on a computer.
Published: 2007

10. Computing the principal eigenvalue of the Laplace operator by a stochastic method

Author: Antoine Lejay, Sylvain Maire, Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Modélisation Numérique et Couplages (MNC), Université de Toulon (UTLN), and Partially supported by the Groupe de Recherche MOMAS
Subjects: ACM: G.: Mathematics of Computing/G.3: PROBABILITY AND STATISTICS/G.3.7: Probabilistic algorithms (including Monte Carlo), General Computer Science, Monte Carlo method, 010103 numerical & computational mathematics, First eigenvalue of the Dirichlet problem, 01 natural sciences, Theoretical Computer Science, 010104 statistics & probability, symbols.namesake, ACM: G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS/G.1.8: Partial Differential Equations, Robustness (computer science), random walk on spheres, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Eigenvalues and eigenvectors, Brownian motion, Mathematics, Numerical Analysis, Euler scheme for Brownian motion, Dirichlet conditions, Applied Mathematics, Mathematical analysis, random walk on rectangles, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Modeling and Simulation, Bounded function, symbols, p-Laplacian, AMS 65C05, 60F15, Laplace operator, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: http://www.elsevier.com/wps/find/journaldescription.cws_home/505615/description#description; International audience; We describe a Monte Carlo method for the numerical computation of the principal eigenvalue of the Laplace operator in a bounded domain with Dirichlet conditions. It is based on the estimation of the speed of absorption of the Brownian motion by the boundary of the domain. Various tools of statistical estimation and different simulation schemes are developed to optimize the method. Numerical examples are studied to check the accuracy and the robustness of our approach.
Published: 2007

11. A Donsker theorem to simulate one-dimensional processes with measurable coefficients

Author: Pierre Etore, Antoine Lejay, Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), and Supported by the Groupe de Recherche MOMAS
Subjects: Statistics and Probability, Donsker theorem, 010102 general mathematics, Monte Carlo method, Mathematical analysis, Feynman–Kac formula, Monte Carlo methods, Feynman-Kac formula, Random walk, Grid, 01 natural sciences, AMS Classification: 60J60, 65C, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Operator (computer programming), one-dimensional process, divergence form operators, scale function, State space, elliptic PDE problem, 0101 mathematics, Donsker's theorem, Empirical process, Mathematics
Abstract: International audience; In this paper, we prove a Donker theorem for one-dimensional processes generated by an operator with measurable coefficients. We construct a random walk on any grid on the state space, using the transition probabilities of the approximated process, and the conditional average times it spends on each cell of the grid. Indeed we can compute these quantities by solving some suitable elliptic PDE problems.
Published: 2007

12. The 1-d stochastic wave equation driven by a fractional Brownian motion

Author: Lluís Quer-Sardanyons, Samy Tindel, Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Statistics and Probability, Holder exponent, Fractional Brownian motion, Stochastic process, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Probability (math.PR), 60H15, 60G15, 35L05, Young integration, Wave equation, 01 natural sciences, Signal, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Dimension (vector space), Mathematics::Probability, Modelling and Simulation, Modeling and Simulation, FOS: Mathematics, 0101 mathematics, Mathematics - Probability, Mathematics
Abstract: In this paper, we develop a Young integration theory in dimension 2 which will allow us to solve a non-linear one dimensional wave equation driven by an arbitrary signal whose rectangular increments satisfy some H\"{o}lder regularity conditions, for some H\"older exponent greater than 1/2. This result will be applied to the infinite dimensional fractional Brownian motion., Comment: 37 pages, 3 figures
Published: 2007

13. APPROXIMATION OF ONE-DIMENSIONAL DIFFUSION PROCESSES WITH DISCONTINUOUS COEFFICIENTS AND APPLICATIONS TO SIMULATION

Author: Etore, Pierre, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Université Henri Poincaré - Nancy 1, Bernard Roynette(bernard.roynette@iecn.u-nancy.fr), projet OMEGA, and GdR MOMAS
Subjects: Donsker theorem, schéma numériques, marche aléatoire, diffusions en dimension un, EDP elliptiques, Feynman-Kac formula, elliptic PDE, one-dimensional diffusion processes, EDS avec temps local, random walk, SDE with local time, formule de Feynman-Kac, [MATH]Mathematics [math], numeric schemes, Skew Brownian Motion, théorème de Donsker
Abstract: In this thesis numerical schemes for processes /X/ generated byoperators with discontinuous coefficients are studied.A first scheme for the one-dimensional case uses DifferentialStochastic Equations with Local Time. Indeed, in dimension one, theprocesses /X/ are solutions of such equations. We construct a grid onthe real line, that is transformed by a proper bijection in a uniformgrid of step /h/. This bijection also transforms /X/ in some process/Y/, that behaves locally like a Skew Brownian Motion (SBM). We knowthe transition probabilities of the SBM on a uniform grid, and theaverage time it spends on each of its cells. A random walk can thenbe built, that converges to /X/ in square root of /h/. A secondscheme, that is more general, is proposed still for the dimensionone. A non uniform grid on the real line is given, whose cells have asize proportional to /h/. Both the transition probabilities of /X/ onthis grid, and the average time it spends on each of its cells, canbe related to the solutions of proper elliptic PDE problems, usingthe Feynman-Kac formula. A time-space random walk can then be built,that converges to /X/ again in square root of /h/. Next somedirections to adapt this approach to the two-dimensional case aregiven. Finally numerical exemples illustrate the studied schemes.; Dans cette thèse on étudie des schémas numériques pour des processus/X/ à coefficients discontinus. Un premier schéma pour le casunidimensionnel utilise les Équations Différentielles Stochastiquesavec Temps Local. En effet en dimension un les processus /X/ sontsolutions de telles équations. On construit une grille sur la droiteréelle, qu'une bijection adéquate transforme en une grille uniformede pas /h/. Cette bijection permet de transformer /X/ en /Y/ qui secomporte localement comme un Skew Brownian Motion, pour lequel onconnaît les probabilités de transition sur une grille uniforme, et letemps moyen passé sur chaque cellule de cette grille. Une marchealéatoire peut alors être construite, qui converge vers /X/ en racinede /h/. Toujours dans le cas unidimensionnel on propose un deuxièmeschéma plus général. On se donne une grille non uniforme sur ladroite réelle, dont les cellules ont une taille proportionnelle à/h/. On montre qu'on peut relier les probabilités de transition de/X/ sur cette grille, ainsi que le temps moyen passé par /X/ surchacune de ses cellules, à des solutions de problèmes d'EDPelliptiques ad hoc. Une marche aléatoire en temps et en espace estainsi construite, qui permet d'approcher /X/ à nouveau en racine de/h/. Ensuite on présente des pistes pour adapter cette dernièreapproche au cas bidimensionnel et les problèmes que cela soulève.Enfin on illustre par des exemples numériques les schémas étudiés.
Published: 2006

14. APPROXIMATION DE PROCESSUS DE DIFFUSION À COEFFICIENTS DISCONTINUS EN DIMENSION UN ET APPLICATIONS À LA SIMULATION

Author: Etoré, Pierre, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Université Henri Poincaré - Nancy 1, Bernard Roynette(bernard.roynette@iecn.u-nancy.fr), projet OMEGA, and GdR MOMAS
Subjects: Donsker theorem, schéma numériques, marche aléatoire, diffusions en dimension un, EDP elliptiques, Feynman-Kac formula, elliptic PDE, one-dimensional diffusion processes, EDS avec temps local, random walk, SDE with local time, formule de Feynman-Kac, [MATH]Mathematics [math], numeric schemes, Skew Brownian Motion, théorème de Donsker
Abstract: In this thesis numerical schemes for processes /X/ generated byoperators with discontinuous coefficients are studied.A first scheme for the one-dimensional case uses DifferentialStochastic Equations with Local Time. Indeed, in dimension one, theprocesses /X/ are solutions of such equations. We construct a grid onthe real line, that is transformed by a proper bijection in a uniformgrid of step /h/. This bijection also transforms /X/ in some process/Y/, that behaves locally like a Skew Brownian Motion (SBM). We knowthe transition probabilities of the SBM on a uniform grid, and theaverage time it spends on each of its cells. A random walk can thenbe built, that converges to /X/ in square root of /h/. A secondscheme, that is more general, is proposed still for the dimensionone. A non uniform grid on the real line is given, whose cells have asize proportional to /h/. Both the transition probabilities of /X/ onthis grid, and the average time it spends on each of its cells, canbe related to the solutions of proper elliptic PDE problems, usingthe Feynman-Kac formula. A time-space random walk can then be built,that converges to /X/ again in square root of /h/. Next somedirections to adapt this approach to the two-dimensional case aregiven. Finally numerical exemples illustrate the studied schemes.; Dans cette thèse on étudie des schémas numériques pour des processus/X/ à coefficients discontinus. Un premier schéma pour le casunidimensionnel utilise les Équations Différentielles Stochastiquesavec Temps Local. En effet en dimension un les processus /X/ sontsolutions de telles équations. On construit une grille sur la droiteréelle, qu'une bijection adéquate transforme en une grille uniformede pas /h/. Cette bijection permet de transformer /X/ en /Y/ qui secomporte localement comme un Skew Brownian Motion, pour lequel onconnaît les probabilités de transition sur une grille uniforme, et letemps moyen passé sur chaque cellule de cette grille. Une marchealéatoire peut alors être construite, qui converge vers /X/ en racinede /h/. Toujours dans le cas unidimensionnel on propose un deuxièmeschéma plus général. On se donne une grille non uniforme sur ladroite réelle, dont les cellules ont une taille proportionnelle à/h/. On montre qu'on peut relier les probabilités de transition de/X/ sur cette grille, ainsi que le temps moyen passé par /X/ surchacune de ses cellules, à des solutions de problèmes d'EDPelliptiques ad hoc. Une marche aléatoire en temps et en espace estainsi construite, qui permet d'approcher /X/ à nouveau en racine de/h/. Ensuite on présente des pistes pour adapter cette dernièreapproche au cas bidimensionnel et les problèmes que cela soulève.Enfin on illustre par des exemples numériques les schémas étudiés.
Published: 2006

15. A Fault Tolerant and Multi-Paradigm Grid Architecture for Time Constrained Problems. Application to Financial Option Pricing

Author: Bezinne, Sébastien, Galtier, Virginie, Vialle, Stéphane, Baude, Françoise, Bossy, Mireille, Dung, Viet, Henrio, Ludovic, SUPELEC-Campus Metz, Ecole Supérieure d'Electricité - SUPELEC (FRANCE), Active objects, semantics, Internet and security (OASIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-COMmunications, Réseaux, systèmes Embarqués et Distribués (Laboratoire I3S - COMRED), Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Probabilistic numerical methods (OMEGA), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), S. Kawata, Université Nice Sophia Antipolis (1965 - 2019) (UNS), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS)
Subjects: [INFO.INFO-DC]Computer Science [cs]/Distributed, Parallel, and Cluster Computing [cs.DC]
Abstract: International audience; This paper introduces a Grid software architecture offering fault tolerance, dynamic and aggressive load balancing and two complementary parallel programming paradigms. Experiments with financial applications on a real multi-site Grid assess this solution. This architecture has been designed to run industrial and financial applications, that are frequently time constrained and CPU consuming, feature both tightly and loosely coupled parallelism requiring generic programming paradigm, and adopt client-server business architecture.
Published: 2006

16. On a Monte Carlo method for neutron transport criticality computations

Author: Sylvain Maire, Denis Talay, TOSCA, Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria), Modélisation Numérique et Couplages (MNC), Université de Toulon (UTLN), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Neutron transport, Applied Mathematics, General Mathematics, Numerical analysis, 010102 general mathematics, Monte Carlo method, 01 natural sciences, Integral equation, 010101 applied mathematics, Hybrid Monte Carlo, Computational Mathematics, Criticality, [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, Dynamic Monte Carlo method, Statistical physics, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Monte Carlo molecular modeling, Mathematics
Abstract: We give a stochastic representation of the principal eigenvalue of some homogeneous neutron transport operators. Our construction is based upon the Feynman-Kac formula for integral transport equations, and uses probabilistic techniques only. We develop a Monte Carlo method for criticality computations. We numerically test this method on various homogeneous and inhomogeneous problems, and compare our results with those obtained by standard methods.
Published: 2006

17. A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients

Author: Miguel Martinez, Antoine Lejay, Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Groupe de recherche MOMAS, and INRIA
Subjects: Statistics and Probability, one-dimensional diffusion, One dimensional diffusion, speed measure, 01 natural sciences, 010104 statistics & probability, local time, FOS: Mathematics, Infinitesimal generator, 0101 mathematics, Diffusion (business), Brownian motion, Mathematics, 60J60, 65C, 60J60 (Primary) 65C (Secondary), skew Brownian motion, Probability (math.PR), 010102 general mathematics, Mathematical analysis, Skew, Monte Carlo methods, 16. Peace & justice, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Discontinuity (linguistics), Scheme (mathematics), scale function, Statistics, Probability and Uncertainty, Mathematics - Probability, Smoothing, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], divergence form operator
Abstract: The aim of this article is to provide a scheme for simulating diffusion processes evolving in one-dimensional discontinuous media. This scheme does not rely on smoothing the coefficients that appear in the infinitesimal generator of the diffusion processes, but uses instead an exact description of the behavior of their trajectories when they reach the points of discontinuity. This description is supplied with the local comparison of the trajectories of the diffusion processes with those of a skew Brownian motion., Comment: Published at http://dx.doi.org/10.1214/105051605000000656 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
Published: 2006

18. On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients

Author: Pierre Etore, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), and GdR MOMAS
Subjects: Statistics and Probability, Heterogeneous random walk in one dimension, 010102 general mathematics, Mathematical analysis, Skew, Monte Carlo methods, Random walk, Heavy traffic approximation, 01 natural sciences, random walk, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Diffusion process, Mathematics::Probability, one-dimensional process, Diffusion-limited aggregation, 0101 mathematics, Statistics, Probability and Uncertainty, Skew Brownian motion, Divergence (statistics), Brownian motion, 60J60, 65C, Mathematics, divergence form operator
Abstract: In this paper, we provide a scheme for simulating one-dimensional processes generated by divergence or non-divergence form operators with discontinuous coefficients. We use a space bijection to transform such a process in another one that behaves locally like a Skew Brownian motion. Indeed the behavior of the Skew Brownian motion can easily be approached by an asymmetric random walk.
Published: 2006

19. Young integrals and SPDEs

Author: Antoine Lejay, Massimiliano Gubinelli, Samy Tindel, Dipartimento di Matematica Applicata [Pisa] (DMA), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS)
Subjects: Pure mathematics, Function space, AMS 60H15, 60H05, fractional Brownian motion, Type (model theory), [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, stochastic heat equation, Potential theory, 47D62, 010104 statistics & probability, Mathematics::Probability, FOS: Mathematics, pathwise stochastic integration, 0101 mathematics, Mathematics, Discrete mathematics, 60H15, Continuous function (set theory), Generator (category theory), Semigroup, Probability (math.PR), 010102 general mathematics, Lipschitz continuity, Functional Analysis (math.FA), Mathematics - Functional Analysis, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Distribution (mathematics), rough path theory, Mathematics - Probability, Analysis
Abstract: In this note, we study the non-linear evolution problem $dY_t = -A Y_t dt + B(Y_t) dX_t$, where $X$ is a $\gamma$-H\"older continuous function of the time parameter, with values in a distribution space, and $-A$ the generator of an analytical semigroup. Then, we will give some sharp conditions on $X$ in order to solve the above equation in a function space, first in the linear case (for any value of $\gamma$ in $(0,1)$), and then when $B$ satisfies some Lipschitz type conditions (for $\gamma>1/2$). The solution of the evolution problem will be understood in the mild sense, and the integrals involved in that definition will be of Young type., Comment: 22 pages, no figures
Published: 2006

20. Using game theory for the electricity market

Author: Bossy, Mireille, Maïzi, Nadia, Jan Olsder, Geert, Pourtallier, Odile, Tanré, Etienne, Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Centre de Mathématiques Appliquées (CMA), Mines Paris - PSL (École nationale supérieure des mines de Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Delft Institute of Applied Mathematics (DIAM), Delft University of Technology (TU Delft), Constraints solving, optimization and robust interval analysis (COPRIN), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-École des Ponts ParisTech (ENPC), INRIA, and MINES ParisTech - École nationale supérieure des mines de Paris
Subjects: [INFO.INFO-OH]Computer Science [cs]/Other [cs.OH], SPOT MARKETS, ELECTRICITY MARKET, NASH EQUILIBRIUM, GAME THEORY
Abstract: We consider a static game model to describe a spot electricity market in which $\calS$ suppliers offer electricity. Each supplier submit a price function to the market, to which the market reacts by fixing the quantities bought to each supplier. The objective of the market is to satisfy its fixed demand, whenever possible, at minimal price. We investigate two cases. In the first case, each of the suppliers strives to maximize its market share on the market; in the second case each supplier strives to maximize its profit. We also make some remarks when suppliers may bring electricity on several markets.
Published: 2006

21. Numerical Time Schemes for an Ocean Related System of PDEs

Author: Petcu, Madalina, Rousseau, Antoine, Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS)
Subjects: [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: International audience; In this article we consider a system of equations related to the delta–Primitive Equations of the ocean and the atmosphere, linearized around a stratified flow, and we supplement the equations with transparent boundary conditions. We study the stability of different numerical schemes and we show that for each case, letting the vertical viscosity delta go to 0, the stability conditions are the same as the classical CFL conditions for the transport equation.
Published: 2006

22. Stochastic Differential Equations Driven by Processes Generated by Divergence Form Operators I: A Wong-Zakai Theorem

Author: Antoine Lejay, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), and Partial support from European Union's TMR Stochastic Analysis Network (grant reference 960075)
Subjects: Statistics and Probability, stochastic process generated by divergence-form operators, approximation of trajectories, Continuous-time stochastic process, Differential equation, Stochastic process, 010102 general mathematics, Mathematical analysis, AMS 60H10, 60J60, 01 natural sciences, stochastic differential equations, Dirichlet process, Stochastic partial differential equation, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Stochastic differential equation, Mathematics::Probability, Rough paths, Piecewise, 0101 mathematics, Divergence (statistics), Mathematics
Abstract: http://www.edpsciences.org/journal/index.cfm?edpsname=ps; We show in this article how the theory of "rough paths" allows us to construct solutions of differential equations (SDEs) driven by processes generated by divergence-form operators. For that, we use approximations of the trajectories of the stochastic process by piecewise smooth paths. A result of type Wong-Zakai follows immediately.
Published: 2006

23. On the Constructions of the Skew Brownian Motion

Author: Antoine Lejay, Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), and The author has been supported by the Groupe de Recherche MOMAS.
Subjects: Statistics and Probability, PDE with transmission condition, 01 natural sciences, 010104 statistics & probability, 60J60 (Primary) 60H10, 60J55 (Secondary), FOS: Mathematics, Statistical physics, mathematical modelling, 0101 mathematics, Brownian motion, Mathematics, 60J60, PDE with singular drift, Computer simulation, scale function and speed measure, skew Brownian motion, 010102 general mathematics, Probability (math.PR), Skew, Process (computing), Monte Carlo methods, Construct (python library), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], SDE with local time, excursions of Brownian motion, 60J55, 60H10, Mathematics - Probability
Abstract: This article summarizes the various ways one may use to construct the Skew Brownian motion, and shows their connections. Recent applications of this process in modelling and numerical simulation motivates this survey. This article ends with a brief account of related results, extensions and applications of the Skew Brownian motion., Comment: Published at http://dx.doi.org/10.1214/154957807000000013 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org)
Published: 2006

24. Technical analysis compared to mathematical models under misspecification

Author: De Saporta, Benoîte, Blanchet-Scalliet, Christophette, Tanré, Etienne, Talay, Denis, de Saporta, Benoîte, Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Groupe de Recherche en Economie Théorique et Appliquée (GREThA), Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Quality control and dynamic reliability (CQFD), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), TOSCA, INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche en Informatique et en Automatique (Inria), Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-INRIA Lorraine
Subjects: [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [MATH.MATH-PR] Mathematics [math]/Probability [math.PR]
Published: 2006

25. Limiting laws associated with Brownian motion perturbated by normalized exponential weights I

Author: Roynette, Bernard, Vallois, Pierre, Yor, Marc, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Probabilités et Modèles Aléatoires (LPMA), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
Subjects: [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Sturm-Liouville equation, Ray-Knight's theorems, 60F10, 60F17, 60G44, 60J25, 60J35, 60J55, 60J57, 60J60, 60J65, normalized exponential weights, Bessel processes, rate of convergence
Abstract: We determine the rate of decay of the expectation Z(t) of some multiplicative functional related to Brownian motion up to time t. This permits to prove that the Wiener measure, penalized by this multiplicative functional, converges as t goes to infinity to a probability measure (p.m.) . We obtain the law of the canonical process under this new p.m.
Published: 2006

26. On (p,q)-rough paths

Author: Nicolas Victoir, Antoine Lejay, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Mathematical Institute [Oxford] (MI), University of Oxford [Oxford], and University of Oxford
Subjects: Work (thermodynamics), Rough path, Pure mathematics, controlled differential equation, Hölder continuous paths, Itô and Stratonovich integrals for semi-martingales, rough path, Differential equation, Applied Mathematics, 010102 general mathematics, 01 natural sciences, Combinatorics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Itô and Stratonovich integrals for semi-martingale, [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], Product (mathematics), 0101 mathematics, Itō's lemma, AMS 60H10, 34F05, Analysis, Mathematics
Abstract: We extend the work of T. Lyons [T.J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (2) (1998) 215–310] and T. Lyons and Z. Qian [T. Lyons, Z. Qian, System Control and Rough Paths, Oxford Math. Monogr. Oxford Univ. Press, Oxford, 2002] to deﬁne integrals and solutions of differential equations along product of p and q rough paths, with 1/p +1/q >1. We use this to write an Itô formula at the level of rough paths, and to see that any rough path can always be interpreted as a product of a p-geometric rough path and a p/2-geometric rough path.
Published: 2006

27. Limiting laws for long Brownian Bridges perturbed by their one-sided maximum, III

Author: Marc Yor, Bernard Roynette, Pierre Vallois, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Probabilités et Modèles Aléatoires (LPMA), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Statistics::Theory, General Mathematics, 010102 general mathematics, Probability (math.PR), Limiting, 16. Peace & justice, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Statistics::Machine Learning, Mathematics::Probability, One sided, Local time, Convergence (routing), Calculus, FOS: Mathematics, Statistics::Methodology, 0101 mathematics, Brownian motion, Mathematics - Probability, Mathematics, Mathematical physics
Abstract: Results of penalization of a one-dimensional Brownian motion ]]> ]]> ]]> ]]> ]]> ]]> ]]> (X_t) $, by its one-sided maximum $\big(S_t=\sup_{0 \leq u \leq t}X_u\big)$, which were recently obtained by the authors are improved with the consideration - in the present paper - of the asymptotic behaviour of the likewise penalized Brownian bridges of length $t$, as $t\rightarrow \infty$, or penalizations by functions of $(S_t,X_t)$, and also the study of the speed of convergence, as $t\rightarrow \infty$, of the penalized distributions at time $t$.
Published: 2005

28. Semi-martingales and rough paths theory

Author: Antoine Lejay, Laure Coutin, Laboratoire de Statistique et Probabilités (LSP), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA), and Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)
Subjects: Statistics and Probability, Pure mathematics, Wong-Zakai theorem, Differential equation, Semi-martingales, 010102 general mathematics, Mathematical analysis, $p$-variation, p-variation, Type (model theory), 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Mathematics::Probability, Iterated integrals, 60F17, 60H05, conditions UT and UCV, iterated integrals, 0101 mathematics, Statistics, Probability and Uncertainty, P-variation, rough paths, Mathematics
Abstract: http://www.imstat.org/; We prove that the theory of rough paths, which is used to define path-wise integrals and path-wise differential equations, can be used with continuous semi-martingales. We provide then an almost sure theorem of type Wong-Zakai. Moreover, we show that the conditions UT and UCV, used to prove that one can interchange limits and Ito or Stratonovich integrals, provide the same result when one uses the rough paths theory.
Published: 2005

29. On first range times of linear diffusions

Author: Pierre Vallois, Paavo Salminen, Mathematical department [Abo Finland] (MD), Åbo Academy University, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS)
Subjects: Statistics and Probability, General Mathematics, Bessel bridges, supremum, 01 natural sciences, Combinatorics, 010104 statistics & probability, symbols.namesake, h-transforms, Range (statistics), 0101 mathematics, Brownian motion, Mathematics, First exit time, 010102 general mathematics, Mathematical analysis, Infimum and supremum, Bessel functions, infimum, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Local time, symbols, Interval (graph theory), Statistics, Probability and Uncertainty, Ray-Knight theorem, Random variable, Bessel function
Abstract: In this paper we consider first range times (with randomised range level) of a linear diffusion on R. Inspired by the observation that the exponentially randomised range time has the same law as a similarly randomised first exit time from an interval, we study a large family of non-negative 2-dimensional random variables (X,X′) with this property. The defining feature of the family is F c (x,y)=F c (x+y,0), ∀ x ≥ 0, y ≥ 0, where F c (x,y):=P (X > x, X′ > y) We also explain the Markovian structure of the Brownian local time process when stopped at an exponentially randomised first range time. It is seen that squared Bessel processes with drift are serving hereby as a Markovian element.
Published: 2005

30. Convergence and Stability of a Discontinuous Galerkin Time-Domain method for the 3D heterogeneous Maxwell equations on unstructured meshes

Author: Stephanie Lohrengel, Stéphane Lanteri, Serge Piperno, Loula Fezoui, Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique (CERMICS), Institut National de Recherche en Informatique et en Automatique (Inria)-École des Ponts ParisTech (ENPC), Scientific computing, modeling and numerical analysis (CAIMAN), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Numerical Analysis, Applied Mathematics, Numerical analysis, Surface integral, Mathematical analysis, Basis function, 010103 numerical & computational mathematics, 01 natural sciences, 010101 applied mathematics, Computational Mathematics, symbols.namesake, Maxwell's equations, Discontinuous Galerkin method, Electromagnetism, Modeling and Simulation, symbols, Boundary value problem, 0101 mathematics, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Numerical stability, Mathematics
Abstract: A Discontinuous Galerkin method is used for to the numerical solution of the time-domain Maxwell equations on unstructured meshes. The method relies on the choice of local basis functions, a centered mean approximation for the surface integrals and a second-order leap-frog scheme for advancing in time. The method is proved to be stable for cases with either metallic or absorbing boundary conditions, for a large class of basis functions. A discrete analog of the electromagnetic energy is conserved for metallic cavities. Convergence is proved for Discontinuous elements on tetrahedral meshes, as well as a discrete divergence preservation property. Promising numerical examples with low-order elements show the potential of the method.
Published: 2005

31. INTERPRETATIONS PROBABILISTES D'OPERATEURS SOUS FORME DIVERGENCE ET ANALYSE DE METHODES NUMERIQUES ASSOCIEES

Author: Martinez, Miguel, Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Université de Provence - Aix-Marseille I, and TALAY DENIS(Denis.Talay@sophia.inria.fr)
Subjects: Stochastic Differential Equations, methodes de Monte-Carlo, local time, Schema d'Euler, Divergence Form Operators, Operateurs sous forme divergence, Equations Differentielles Stochastiques, Monte-Carlo methods, [MATH]Mathematics [math], Euler Scheme, Skew Brownian Motion
Abstract: LEPINGLE Dominique, VERETENNIKOV A.Yu, PARDOUX Etienne, PAGES Gilles; The analysis and approximation of soutions of Stochastic Differential Equations (S.D.E.) having discontinuous coefficients is a subject that has not yet been given a satisfactory treatment. This problem becomes particulary motivating when solutions of certain Partial Differential Equations (P.D.E.) that also involve discontinuous coefficients, are beeing approximated using Monte-Carlo methods. This is for example the case, well-known in Physics, of P.D.Es involving a Divergence Form Operator (D.F.O.) with discontinuous coefficients : discontinuities are then closely related to the media where evolves the system under study. This thesis gives new results for the analysis and approximation of solutions of S.D.Es that are related to a D.F.O. with discontinuous coefficients. Statistical aspects of the models in play are also studied.; L'analyse et l'approximation de solutions des Equations Differentielles Stochastiques (E.D.S.) possédant des coefficients discontinus est un sujet qui n'a pas ete traité de facon pleinement satisfaisante. Ce problème devient particulierement motivant lorsque l'on cherche à approcher, par des méthodes de Monte-Carlo, les solutions de certaines Equations aux Derivées Partielles (E.D.P) qui font également intervenir des coefficients discontinus. C'est par exemple le cas, bien connu en Physique, des E.D.P.s avec opérateur sous forme divergence (O.F.D.) dont les coefficients sont discontinus et que nous étudions dans ce mémoire : les discontinuités traduisent alors les irrégularités du milieu dans lequel évolue le système étudié. Cette thèse propose de nouveaux résultats pour l'analyse et l'approximation de solutions des E.D.S. qui sont reliées à un O.F.D. dont les coefficients sont discontinus. Les aspects statistiques des modèles en jeu sont également étudiés.
Published: 2004

32. Approximation of the distribution of the supremum of a centered random walk. Application to the local score

Author: Pierre Vallois, Marie-Pierre Etienne, Laboratoire Statistique et Génome (SG), Institut National de la Recherche Agronomique (INRA)-Université d'Évry-Val-d'Essonne (UEVE)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS)
Subjects: Statistics and Probability, Distribution (number theory), General Mathematics, one-sided maximum, 01 natural sciences, Combinatorics, random walk, 010104 statistics & probability, 03 medical and health sciences, Mathematics::Probability, 60F05, 60F17, 60G17, 60G40, 60G50,60J65, Applied mathematics, Limit (mathematics), 0101 mathematics, 030304 developmental biology, Mathematics, 0303 health sciences, Heterogeneous random walk in one dimension, local score, Loop-erased random walk, Random walk, Infimum and supremum, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Distribution function, Rate of convergence, Skorokhod's embedding
Abstract: International audience; We determine the rate of convergence of the distribution function of the one-sided supremum of a centered random walk to its limit.
Published: 2004

33. Matrices aléatoires: Statistique asymptotique des valeurs propres

Author: Pastur, Leonid, Lejay, Antoine, Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), J. Azéma, M. Émery, M. Ledoux, and M. Yor
Subjects: ensemble gaussien unitaire, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], loi du demi-cercle, ensemble gaussien orthogonal, AMS 15A52 (62E20, 62H05, 82B44), matrices aléatoires, comportement asymptotique du spectre
Abstract: Cet article correspond à un cours donné par L. Pastur dans le cadre de l'École Doctorale de l'Université de Provence en février et mars 2000.; Ce cours est une brève introduction à la théorie de la distribution asymptotique des valeurs propres des matrices aléatoires symétriques réelles ou hermitiennes de grande taille. Cette théorie a connu récemment beaucoup d'évolutions motivées par diverses branches des mathématiques et de la physique. L'étude de quelques régimes asymptotiques mène à des résultats et à des techniques intéressants.
Published: 2004

34. Monte Carlo methods for fissured porous media: a gridless approach

Author: Antoine Lejay, Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Supported by the GdR MOMAS, and K.K. Sabelfeld
Subjects: Statistics and Probability, ACM: G.: Mathematics of Computing/G.3: PROBABILITY AND STATISTICS/G.3.7: Probabilistic algorithms (including Monte Carlo), Darcy's law, diffusion on a graph, Discretization, Computer science, double porosity model, Applied Mathematics, 010102 general mathematics, Monte Carlo method, Darcy law, simulation of stochastic process, [PHYS.PHYS.PHYS-GEO-PH]Physics [physics]/Physics [physics]/Geophysics [physics.geo-ph], 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Applied mathematics, 0101 mathematics, Skew Brownian motion, AMS 76S05 (65C05, 76M25, 76M35), Porous medium, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: In this article, we present two Monte Carlo methods to solve some problems related to the Darcy law in geophysics. Both methods do not require any discretization and are exact methods.
Published: 2003

35. Simulating a diffusion on a graph. Application to reservoir engineering

Author: Antoine Lejay, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), and Partially supported by the Groupe de Recherche MOMAS
Subjects: Statistics and Probability, Discrete mathematics, ACM: G.: Mathematics of Computing/G.3: PROBABILITY AND STATISTICS/G.3.7: Probabilistic algorithms (including Monte Carlo), Constant coefficients, Applied Mathematics, 010102 general mathematics, Monte Carlo method, Constant speed, Monte Carlo methods, one-dimensional diffusion processes, 01 natural sciences, Graph, AMS : 65C, 60K40, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, skew brownian motion, Reservoir engineering, Dynamic Monte Carlo method, Piecewise, Applied mathematics, Infinitesimal generator, 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics, diffusion on graphs
Abstract: We develop a simple Monte Carlo method to compute the position at a given time of a diffusion on a graph, with constant speed on each edge. This method is exact, and we claim it could be used for simulating the position of a particle in a fissured media. Besides, we advocate that the notion of diffusions on graphs could be useful to understand the behavior of one-dimensional diffusions whose infinitesimal generator has piecewise constant coefficients.
Published: 2003

36. On the convergence of stochastic integrals driven by processes converging on account of a homogenization property

Author: Antoine Lejay, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), and TMR Stochastic Analysis
Subjects: Statistics and Probability, Continuous-time stochastic process, AMS: 60F17, 60K40, Stochastic calculus, homogenization, 01 natural sciences, Homogenization (chemistry), Levy area, Stochastic integral, 010104 statistics & probability, Stochastic differential equation, Applied mathematics, 0101 mathematics, Mathematics, Stochastic process, Lévy area, 010102 general mathematics, Mathematical analysis, stochastic differential equations, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], good sequence of semimartingales, 60F17, conditions UT and UCV, Statistics, Probability and Uncertainty, Counterexample
Abstract: http://www.math.washington.edu/~ejpecp/; International audience; We study the limit of functionals of stochastic processes for which an homogenization result holds. All these functionals involve stochastic integrals. Among them, we consider more particularly the Lévy area and those giving the solutions of some SDEs. The main question is to know whether or not the limit of the stochastic integrals is equal to the stochastic integral of the limit of each of its terms. In fact, the answer may be negative, especially in presence of a highly oscillating first-order differential term. This provides us some counterexamples to the theory of good sequence of semimartingales.
Published: 2003

37. On the Importance of the Levy Area for Studying the Limits of Functions of Converging Stochastic Processes. Application to Homogenization

Author: Lejay, Antoine, Lyons, Terry, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Mathematical Institute [Oxford] (MI), University of Oxford [Oxford], Réseau Européen TMR Stochastic Analysis (Project ERBF MRX CT 960075A, grant reference 960075), D. Bakry, L. Beznea, Gh. Bucur, M. Röckner, and University of Oxford
Subjects: [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], approximation of SDEs by ODEs, homogenization, p-variation, AMS 60J60 (secondary) 35B27, 60F17, rough paths, Lévy area: convergence of stochastic processes
Abstract: Two concrete examples show us that the convergence of a family of stochastic processes "as controls", i.e. as integrators of SDEs or differential forms, may require more information than simply the limit in the uniform norm of the processes. This may be particularly important when one deals with the homogenization theory. The theory of rough paths is then used to bring some new results about interchanging limits and functionals of stochastic processes.
Published: 2003

38. An Introduction to Rough Paths

Author: Antoine Lejay, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), J. Azéma, M. Ledoux, M. Emery, and M. Yor
Subjects: controlled differential equations, Pure mathematics, Rough path, Differential form, Differential equation, Stochastic process, 010102 general mathematics, Mathematical analysis, p-variation, [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], Extension (predicate logic), geometric multiplicative functional, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Order of integration (calculus), Chen series, 010104 statistics & probability, integration against irregular paths, Iterated integrals, iterated integrals, stochastic processes, 0101 mathematics, Brownian motion, Mathematics
Abstract: This article aims to be an introduction to the theory of rough paths, in which integrals of differential forms against irregular paths and differential equations controlled by irregular paths are defined. This theory makes use of an extension of the notion of iterated integrals of the paths, whose algebraic properties appear to be fundamental. This theory is well-suited for stochastic processes.
Published: 2003

39. On the decomposition of excursions measures of processes whose generators have diffusion coefficients discontinuous at one point

Author: Antoine Lejay, Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), This work has been partially supported by European Union's TMR Stochastic Analysis Network (grant reference 960075), and Lejay, Antoine
Subjects: [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Mathematics::Probability, scale function and speed measure, excursions theory, skew Brownian motion, Monte Carlo methods, AMS: 60J60, 60J55
Abstract: The coefficients in the decomposition of the excursions measure as convex combination of excursions measures of reflected processes are computed in order to characterize the discontinuity at one point of the diffusion coefficient. In some sense, this result extends to general diffusions a similar one for the skew Brownian motion, and we advocate it may be used in Monte Carlo methods for discontinuous media.
Published: 2002

40. On independent times and positions for Brownian motions

Author: Marc Yor, Bernard Roynette, Bernard de Meyer, Pierre Vallois, CEntre de Recherche en Mathématiques, Statistique et Économie Mathématique (CERMSEM), Université Paris 1 Panthéon-Sorbonne (UP1)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut Universitaire de France (IUF), and Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.)
Subjects: Bessel process, General Mathematics, Holomorphic function, Hadamard's theorem, 01 natural sciences, Combinatorics, 010104 statistics & probability, Ornstein-Uhlenbeck and Bessel processes, Space-time Brownian motion, 60J25, Stopping time, 60J65, 60G44, 0101 mathematics, 60G40, Brownian motion, ComputingMilieux_MISCELLANEOUS, Mathematics, 010102 general mathematics, Ornstein–Uhlenbeck process, Exponential function, Skorokhod embedding, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Bounded function, 60E10, Constant (mathematics)
Abstract: Let $(B_t ; t \ge 0)$, $\big(\mbox{resp. }((X_t, Y_t) ; t \ge 0)\big)$ be a one (resp. two) dimensional Brownian motion started at 0. Let $T$ be a stopping time such that $(B_{t \wedge T} ; t \ge 0)$ \big(resp. $(X_{t \wedge T} ; t \ge 0) ; (Y_{t \wedge T} ; t \ge 0)\big)$ is uniformly integrable. The main results obtained in the paper are: \begin{itemize} \item[1)] if $T$ and $B_T$ are independent and $T$ has all exponential moments, then $T$ is constant. \item[2)] If $X_T$ and $Y_T$ are independent and have all exponential moments, then $X_T$ and $Y_T$ are Gaussian. \end{itemize} We also give a number of examples of stopping times $T$, with only some exponential moments, such that $T$ and $B_T$ are independent, and similarly for $X_T$ and $Y_T$. We also exhibit bounded non-constant stopping times $T$ such that $X_T$ and $Y_T$ are independent and Gaussian.
Published: 2002

41. Worst Case Management of the Model Risk in Option Hedging

Author: Talay, Denis, Zheng, Ziyu, Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), and INRIA
Subjects: Computer Science::Computer Science and Game Theory, FINANCIAL MATHEMATICS, [INFO.INFO-OH]Computer Science [cs]/Other [cs.OH], MODEL RISK, OPTION HEDGING
Abstract: We are interested in Model Risk control problems. We study a strategy for the trader which, in a sense, guarantees good performances whatever is the unknown model for the assets of his/her portfolio. We consider the Model Risk control problem as a two players (Trader versus Market) zero-sum stochastic differential game problem. corresponding to a `worst case' worry: the trader chooses trading strategies to decrease the risk and therefore acts as a minimizer; the market systematically acts against the interest of the trader, so that we consider it acts as a maximizer. This stochastic game problem can be viewed as a continuous-time extension of discrete-time strategies based upon prescriptions issued from VaR analyses at the beginning of each period. In addition, the initial value of the optimal portfolio can be seen as the minimal amount of money which is needed to face the worst possible damage. We give a proper mathematical statement for such a game problem. We prove that the value function of this game problem is the unique viscosity solution to an Hamilton-Jacobi-B-ell- man-Isaacs equation, and satisfies the Dynamic Programming Principle.
Published: 2002

42. A Stochastic Particle Method with Random Weights for the Computation of Statistical Solutions of McKean-Vlasov Equations. Part II: Convergence Rate of The Method

Author: Talay, Denis, Vaillant, Olivier, Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), and INRIA
Subjects: STATISTICAL SOLUTIONS, [INFO.INFO-OH]Computer Science [cs]/Other [cs.OH], STOCHASTIC PARTICLE SYSTEM, MCKEAN-VLASOV EQUATIONS
Abstract: In the first part of this paper [7], we have proposed a stochastic particle method to compute statistical solutions of a McKean-Vlasov equation with random initial condition and we have empirically studied its convergence rate. In this second part, we estimate the convergence rate of our method in terms of the number of simulated particles and the time discretization step.
Published: 2001

43. Sur l'indépendance d'un temps d'arrêt $T$ et de la position $B_T$ d'un mouvement brownien $(B_u, u\geq 0)$

Author: De Meyer, Bernard, Yor, Marc, Roynette, Bernard, Vallois, Pierre, CEntre de Recherche en Mathématiques, Statistique et Économie Mathématique (CERMSEM), Université Paris 1 Panthéon-Sorbonne (UP1)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC), and Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)
Subjects: [MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
Abstract: International audience; In this Note, we describe many examples of two-dimensional random variables {$B_T, T$} obtained from the position of a Brownian motion $(B_t, t\geq 0)$. at a stopping time $T$ such that $T$ and $B_T$ are independent. For such pairs, the law of $T$ determines that of $B_T$ and vice versa; we study the constraints on these laws induced by the independence assumption.; Dans cette Note, nous présentons de nombreux exemples de variables aléatoires 2-dimensionnelles {$B_T, T$} obtenues en considérant la position d'un mouvement brownien $(B_t, t\geq 0)$. pris en un temps d'arrêt $T$ tel que $T$ et $B_T$ sont indépendants. Pour de tels couples, la loi de $T$ détermine celle de $B_T$, et inversement ; nous étudions les contraintes induites sur ces lois par l'hypothèse d'indépendance.
Published: 2001

44. A Stochastic Particle Method with Random Weights for the Computation of Statistical Solutions of McKean-Vlasov Equations. Part I: Foundation of the Method and Empirical Evidence

Author: Talay, Denis, Vaillant, Olivier, Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), and INRIA
Subjects: Physics::Fluid Dynamics, STATISTICAL SOLUTIONS, [INFO.INFO-OH]Computer Science [cs]/Other [cs.OH], STOCHASTIC PARTICLE SYSTEM, MCKEAN-VLASOV EQUATIONS
Abstract: We are interested in statistical solutions [28] of McKean-Vlasov equations. An example of motivation is the Navier-Stokes equation for the vorticity of a 2D incompressible fluid flow. We propose an original and efficient numerical method to compute moments of such solutions. It is a stochastic particle method with random weights. These weights are defined through nonparametric estimators of a regression function, and convey the uncertainty on the initial condition of the considered equation. In this first part of our work, we prove an existence and uniqueness result for a class of nonlinear stochastic differential equations (SDE), and we study the relation between these nonlinear SDEs and statistical solutions of the corresponding McKean-Vlasov equations. This result founds our stochastic particle method, for which we show results of numerical experiments obtained for the Burgers and the 2D Navier-Stokes equation.
Published: 2001

45. Sojourn time of some reflected Brownian motion in the unit disk

Author: Deaconu, Madalina, Gradinaru, Mihai, Roche, Jean Rodolphe, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS)
Subjects: [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], numerical computations, boundary value problems, fractional linear transformation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 60J65, 60H30, 35K20, 35J25, 30C35, 30E25, Reflected Brownian motion
Abstract: We study the heat diffusion in a domain with an obstacle inside. More precisely, we are interested in the quantity of heat in so far as a function of the position of the heat source at time 0. This quantity is also equal to the expectation of the sojourn time of the Brownian motion, reflected on the boundary of a small disk contained in the unit disk, and killed on the unit circle. We give the explicit expression of this expectation.This allows us to make some numerical estimates and thus to illustrate the behaviour of this expectation as a function of starting point of Brownian motion.
Published: 2000

46. A Monte Carlo Method to Compute the exchange coefficient in the double porosity model

Author: Fabien Campillo, Antoine Lejay, Applications of interacting particle systems to statistics (ASPI), Université de Rennes (UR)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), VSP, Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Inria Rennes – Bretagne Atlantique, Stochastic Dynamical Systems (SYSDYS), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
Subjects: Statistics and Probability, Monte Carlo method, [PHYS.PHYS.PHYS-GEO-PH]Physics [physics]/Physics [physics]/Geophysics [physics.geo-ph], 01 natural sciences, 010305 fluids & plasmas, Physics::Geophysics, Hybrid Monte Carlo, 010104 statistics & probability, 0103 physical sciences, Statistics, Kinetic Monte Carlo, Statistical physics, 0101 mathematics, Mathematics, double porosity model, Applied Mathematics, Monte Carlo methods, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], random walk on squares, Macroscopic scale, AMS 76S05, 65C05, 76M35, Dynamic Monte Carlo method, Monte Carlo method in statistical physics, Porous medium, fissured media, Monte Carlo molecular modeling
Abstract: This article was written while the authors were members of the SYSDYS team (INRIA Sophia-Antipolis & LATP, Universite de Provence, Marseille).; The double porosity model allows to compute the pressure at a macroscopic scale in a fractured porous media, but requires the computation of some exchange coefficient characterizing the passage of the fluid from and to the porous media (the matrix) and the fractures. We propose a new Monte Carlo method to estimate this coefficient. Here we give an overview of some article from F. Campillo and A. Lejay
Published: 2000

47. Optimal Rate of Convergence of a Stochastic Particle Method to Solutions of 1D Viscous Scalar Conservation Law Equations

Author: Mireille Bossy, Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), and INRIA
Subjects: STOCHASTIC PARTICLE METHOD, [INFO.INFO-OH]Computer Science [cs]/Other [cs.OH], EULER SCHEME, STOCHASTIC PROCESSES
Abstract: The aim of this work is to present the analysis of the rate of convergence of a stochastic particle method for 1D viscous scalar conservation law equations. The convergence rate result is $\mathcal O(\D t + 1/\sqrt{N})$, where $N$ is the number of numerical particles and $\D t$ is the time step of the first order Euler scheme applied to the dynamic of the interacting particles.
Published: 2000

48. Weak solution of semi-linear PDE, BSDE and homogenization

Author: Antoine Lejay, Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), and 0929-9629
Subjects: Statistics and Probability, divergence-form operators, Applied Mathematics, Weak solution, 010102 general mathematics, AMS 35B27, 35K55, 35R60, 60H15, homogenization, BSDE, 01 natural sciences, Homogenization (chemistry), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, semi-linear PDE, Applied mathematics, 0101 mathematics, Mathematics
Abstract: The sketch of the proof using BSDE of a homogenization result for semi-linear PDE with a divergence-form operator is given here. The method employed here relies on the use of a weak topology.
Published: 2000

49. Optimal Sequencing of Contract Algorithms

Author: Zilberstein, Shlomo, Charpillet, François, Chassaing, Philippe, Department of physics and astronomy (UMASS), University of Massachusetts [Amherst] (UMass Amherst), University of Massachusetts System (UMASS)-University of Massachusetts System (UMASS), Autonomous intelligent machine (MAIA), INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)
Subjects: mathematical foundation, flexible computation, real time, mata-level control, IA, AI, temps reel, [INFO.INFO-OH]Computer Science [cs]/Other [cs.OH], contract algorithms, real-time problem solving, algorithmes par contrat
Abstract: Article dans revue scientifique avec comité de lecture. internationale.; International audience; We address the problem of building an interruptible real-time system using non-interruptible components. Some artificial intelligence techniques offer a tradeoff between computation time and quality of results, but their run-time must be determined when they are activated. These techniques, called contract algorithms, introduce a complex scheduling problem when there is uncertainty about the amount of time available for problem-solving. We show how to optimally sequence contract algorithms to create the best possible interruptible system with or without stochastic information about the deadline. These results extend the foundation of real-time problem-solving and provide useful guidance for embedding contract algorithms in applications.
Published: 1999

50. Comparison of a stochastic particle method and a finite volume deterministic method applied to Burgers equation

Author: Serge Piperno, Loula Fezoui, Mireille Bossy, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), Probabilistic numerical methods (OMEGA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Centre National de la Recherche Scientifique (CNRS), Scientific computing, modeling and numerical analysis (CAIMAN), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Statistics and Probability, Partial differential equation, Finite volume method, Applied Mathematics, Mathematical analysis, Particle method, 010103 numerical & computational mathematics, Finite volume method for one-dimensional steady state diffusion, 16. Peace & justice, 01 natural sciences, Burgers' equation, 010104 statistics & probability, 0101 mathematics, Deterministic method, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract: International audience
Published: 1997

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