The Experts below are selected from a list of 16677 Experts worldwide ranked by ideXlab platform
Jeremie Unterberger - One of the best experts on this subject based on the ideXlab platform.
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from Constructive field theory to fractional stochastic calculus ii Constructive Proof of convergence for the levy area of fractional brownian motion with hurst index alpha in frac 1 8 frac 1 4
Annales Henri Poincaré, 2012Co-Authors: Jacques Magnen, Jeremie UnterbergerAbstract:Let B = (B1(t), . . . ,Bd(t)) be a d-dimensional fractional Brownian motion with Hurst index α < 1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of B is a difficult task because of the low Holder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to B, or to solving differential equations driven by B. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using “standard” tools of Constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates and call for an extension of Gaussian tools such as, for instance, the Malliavin calculus. After a first introductory paper (Magnen and Unterberger in From Constructive theory to fractional stochastic calculus. (I) An introduction: rough path theory and perturbative heuristics, 2011), this one concentrates on the details of the Constructive Proof of convergence for second-order iterated integrals, also known as Levy area. A summary in French may be found in Unterberger (Mode d’emploi de la theorie Constructive des champs bosoniques, avec une application aux chemins rugueux, 2011).
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from Constructive field theory to fractional stochastic calculus ii Constructive Proof of convergence for the l evy area of fractional brownian motion with hurst index alpha in 1 8 1 4
arXiv: Probability, 2011Co-Authors: Jacques Magnen, Jeremie UnterbergerAbstract:{Let $B=(B_1(t),...,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $\alpha<1/4$, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of $B$ is a difficult task because of the low H\"older regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to $B$, or to solving differential equations driven by $B$. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using "standard" tools of Constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates, and call for an extension of Gaussian tools such as for instance the Malliavin calculus. After a first introductory paper \cite{MagUnt1}, this one concentrates on the details of the Constructive Proof of convergence for second-order iterated integrals, also known as L\'evy area.
Tianhao Zhang - One of the best experts on this subject based on the ideXlab platform.
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a Constructive Proof of the cauchy kovalevskaya theorem for ordinary differential equations
Journal of Fixed Point Theory and Applications, 2021Co-Authors: Shane Kepley, Tianhao ZhangAbstract:We give a Constructive Proof of the classical Cauchy–Kovalevskaya theorem for ordinary differential equations which provides a sufficient condition for an initial value problem to have a unique, analytic solution. Our Proof is inspired by a modern numerical technique for rigorously solving nonlinear problems known as the radii polynomial approach. The main idea is to recast the existence and uniqueness of analytic solutions as a fixed point problem on an appropriately chosen Banach space, and then prove a fixed point exists via a Constructive version of the Banach fixed point theorem. A key aspect of this method is the use of an approximate solution which plays a crucial role in the theoretical Proof. Our Proof is Constructive in the sense that we provide an explicit recipe for constructing the fixed point problem, an approximate solution, and the bounds necessary to prove the existence of the fixed point.
Alexey Koloydenko - One of the best experts on this subject based on the ideXlab platform.
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a Constructive Proof of the existence of viterbi processes
IEEE Transactions on Information Theory, 2010Co-Authors: Juri Lember, Alexey KoloydenkoAbstract:Since the early days of digital communication, hidden Markov models (HMMs) have now been also routinely used in speech recognition, processing of natural languages, images, and in bioinformatics. In an HMM (X t,Y t)t ? 1, observations X 1,X 2,... are assumed to be conditionally independent given a Markov process Y 1,Y 2,..., which itself is not observed; moreover, the conditional distribution of X t depends solely on Y t. Central to the theory and applications of HMM is the Viterbi algorithm to find a maximum a posteriori probability (MAP) estimate v(x 1: T)=(v 1,v 2,...,vT) of Y 1: T given observed data x 1: T. Maximum a posteriori paths are also known as the Viterbi paths, or alignments. Recently, attempts have been made to study behavior of the Viterbi alignments when T? ?. Thus, it has been shown that in some cases a well-defined limiting Viterbi alignment exists. While innovative, these attempts have relied on rather strong assumptions and involved Proofs which are existential. This work proves the existence of infinite Viterbi alignments in a more Constructive manner and for a very general class of HMMs.
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a Constructive Proof of the existence of viterbi processes
arXiv: Statistics Theory, 2008Co-Authors: Juri Lember, Alexey KoloydenkoAbstract:Since the early days of digital communication, hidden Markov models (HMMs) have now been also routinely used in speech recognition, processing of natural languages, images, and in bioinformatics. In an HMM $(X_i,Y_i)_{i\ge 1}$, observations $X_1,X_2,...$ are assumed to be conditionally independent given an ``explanatory'' Markov process $Y_1,Y_2,...$, which itself is not observed; moreover, the conditional distribution of $X_i$ depends solely on $Y_i$. Central to the theory and applications of HMM is the Viterbi algorithm to find {\em a maximum a posteriori} (MAP) estimate $q_{1:n}=(q_1,q_2,...,q_n)$ of $Y_{1:n}$ given observed data $x_{1:n}$. Maximum {\em a posteriori} paths are also known as Viterbi paths or alignments. Recently, attempts have been made to study the behavior of Viterbi alignments when $n\to \infty$. Thus, it has been shown that in some special cases a well-defined limiting Viterbi alignment exists. While innovative, these attempts have relied on rather strong assumptions and involved Proofs which are existential. This work proves the existence of infinite Viterbi alignments in a more Constructive manner and for a very general class of HMMs.
Jacques Magnen - One of the best experts on this subject based on the ideXlab platform.
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from Constructive field theory to fractional stochastic calculus ii Constructive Proof of convergence for the levy area of fractional brownian motion with hurst index alpha in frac 1 8 frac 1 4
Annales Henri Poincaré, 2012Co-Authors: Jacques Magnen, Jeremie UnterbergerAbstract:Let B = (B1(t), . . . ,Bd(t)) be a d-dimensional fractional Brownian motion with Hurst index α < 1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of B is a difficult task because of the low Holder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to B, or to solving differential equations driven by B. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using “standard” tools of Constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates and call for an extension of Gaussian tools such as, for instance, the Malliavin calculus. After a first introductory paper (Magnen and Unterberger in From Constructive theory to fractional stochastic calculus. (I) An introduction: rough path theory and perturbative heuristics, 2011), this one concentrates on the details of the Constructive Proof of convergence for second-order iterated integrals, also known as Levy area. A summary in French may be found in Unterberger (Mode d’emploi de la theorie Constructive des champs bosoniques, avec une application aux chemins rugueux, 2011).
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from Constructive field theory to fractional stochastic calculus ii Constructive Proof of convergence for the l evy area of fractional brownian motion with hurst index alpha in 1 8 1 4
arXiv: Probability, 2011Co-Authors: Jacques Magnen, Jeremie UnterbergerAbstract:{Let $B=(B_1(t),...,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $\alpha<1/4$, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of $B$ is a difficult task because of the low H\"older regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to $B$, or to solving differential equations driven by $B$. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using "standard" tools of Constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates, and call for an extension of Gaussian tools such as for instance the Malliavin calculus. After a first introductory paper \cite{MagUnt1}, this one concentrates on the details of the Constructive Proof of convergence for second-order iterated integrals, also known as L\'evy area.
Shane Kepley - One of the best experts on this subject based on the ideXlab platform.
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a Constructive Proof of the cauchy kovalevskaya theorem for ordinary differential equations
Journal of Fixed Point Theory and Applications, 2021Co-Authors: Shane Kepley, Tianhao ZhangAbstract:We give a Constructive Proof of the classical Cauchy–Kovalevskaya theorem for ordinary differential equations which provides a sufficient condition for an initial value problem to have a unique, analytic solution. Our Proof is inspired by a modern numerical technique for rigorously solving nonlinear problems known as the radii polynomial approach. The main idea is to recast the existence and uniqueness of analytic solutions as a fixed point problem on an appropriately chosen Banach space, and then prove a fixed point exists via a Constructive version of the Banach fixed point theorem. A key aspect of this method is the use of an approximate solution which plays a crucial role in the theoretical Proof. Our Proof is Constructive in the sense that we provide an explicit recipe for constructing the fixed point problem, an approximate solution, and the bounds necessary to prove the existence of the fixed point.