arXiv subjects

Nicolas Bouleau

Publications and source records attributed to Nicolas Bouleau.

34 records · Page 2Linked to original sources

arXiv2007

Bringing errors into focus

This lecture presents recent advances in the theory of errors propagation. We first explain in which cases the propagation of errors may be performed with a first order differential calculus or needs a second order differential calculus. Then we point out the link between error propagation and the concept of second order vector in differential geometry, emphasizing the existence of a slight ambiguity concerning the bias operator. The third part in devoted to the powerful framework of Dirichlet forms whose main feature is to apply easily to infinite dimensional models including the Wiener space (giving an interpretation of Malliavin calculus in terms of errors), the Poisson space and the Monte Carlo space. In the fourth part we show how an error in the usual mathematical sense, i.e. an approximate quantity, may yield a Dirichlet form and we introduce the four bias operators. Eventually we connect the Dirichlet form with statistics by identifying the square of field operator with the inverse of the Fisher information matrix.

math.PR↗

arXiv2006

Error structures and parameter estimation

This article proposes a link between statistics and the theory of Dirichlet forms used to compute errors. The error calculus based on Dirichlet forms is an extension of classical Gauss' approach to error propagation. The aim of this paper is to derive error structures from measurements. The links with Fisher's information lay the foundations of a strong connection with experiment. We show that this connection behaves well towards changes of variables and is related to the theory of asymptotic statistics.

math.ST↗

arXiv2006

On some errors related to the graduation of measuring instruments

The error on a real quantity Y due to the graduation of the measuring instrument may be represented, when the graduation is regular and fines down, by a Dirichlet form on R whose square field operator do not depend on the probability law of Y as soon as this law possesses a continuous density. This feature is related to the "arbitrary functions principle" (Poincaré, Hopf). We give extensions of this property to multivariate case and infinite dimensional case for approximations of the Brownian motion. We use a Girsanov theorem for Dirichlet forms which has its own interest. Connections are given with discretization of stochastic differential equations.

math.PR↗

arXiv2006

The Shift, properties and recommendations for practical use

Because the stochastic calculus yields rarely random variables with laws defined by explicit closed formulas, probabilistic numerical computations are done most often by simulation. The simulation by the shift, whose field of application is as wide as that of Monte Carlo method, is particularly relevant when the simulations use, for each sample, a large number of calls to the random function. We give here the theoretical features, the implementation and the specific advantages of this method.

math.PR↗

arXiv2006

On numerical integration by the shift and application to Wiener space

The aim of this study is to clarify the consequences of recent theoretical results for the numerical computation of expectation by the shift method, and in particular to yield sufficient criteria for the existence of speed of convergence of the type `iterated logarithm' in several situations. We deepen the case of the Wiener space because it contains many situations useful in applications.

math.PR↗

arXiv2006

Viscoelasticity and Lévy processes

We show that the linear viscoelastic materials, and more generally the physical phenomena to which Biot's relaxation theory is relevant, can be put in correspondance with the laws of processes with independent increments. In the one dimensional case this correspondence is one to one with subordinators and gives rise naturally to a conjugation relation on subordinators.

math.PR↗

arXiv2006

An extension to the Wiener space of the arbitrary functions principle

The arbitrary functions principle says that the fractional part of $nX$ converges stably to an independent random variable uniformly distributed on the unit interval, as soon as the random variable $X$ possesses a density or a characteristic function vanishing at infinity. We prove a similar property for random variables defined on the Wiener space when the stochastic measure $dB\_s$ is crumpled on itself.

math.PR↗

arXiv2006

Dirichlet forms methods, an application to the propagation of the error due to the Euler scheme

We present recent advances on Dirichlet forms methods either to extend financial models beyond the usual stochastic calculus or to study stochastic models with less classical tools. In this spirit, we interpret the asymptotic error on the solution of an sde due to the Euler scheme in terms of a Dirichlet form on the Wiener space, what allows to propagate this error thanks to functional calculus.

math.PR↗

arXiv2006

Differential calculus for Dirichlet forms : the measure-valued gradient preserved by image

In order to develop a differential calculus for error propagation we study local Dirichlet forms on probability spaces with square field operator $Γ$ -- i.e. error structures -- and we are looking for an object related to $Γ$ which is linear and with a good behaviour by images. For this we introduce a new notion called the measure valued gradient which is a randomized square root of $Γ$. The exposition begins with inspecting some natural notions candidate to solve the problem before proposing the measure-valued gradient and proving its satisfactory properties.

math.PR↗

arXiv2006

Dirichlet forms in simulation

Equipping the probability space with a local Dirichlet form with square field operator $Γ$ and generator $A$ allows to improve Monte Carlo computations of expectations, densities, and conditional expectations, as soon as we are able to simulate a random variable $X$ together with $Γ[X]$ and $A[X]$. We give examples on the Wiener space, on the Poisson space and on the Monte Carlo space. When $X$ is real-valued we give an explicit formula yielding the density at the speed of the law of large numbers.

math.PR↗

arXiv2006

Error calculus and path sensitivity in financial models

In the framework of risk management, for the study of the sensitivity of pricing and hedging in stochastic financial models to changes of parameters and to perturbations of the stock prices, we propose an error calculus which is an extension of the Malliavin calculus based on Dirichlet forms. Although useful also in physics, this error calculus is well adapted to stochastic analysis and seems to be the best practicable in finance. This tool is explained here intuitively and with some simple examples.

math.PR↗

arXiv2006

Calcul d'erreur complet lipschitzien et formes de Dirichlet

We study the error calculus from a mathematical point of view, in particular for the infinite dimensional models met in stochastic analysis. Gauss was the first to propose an error calculus. It can be reinforced by an extension principle based on Dirichlet forms which gives more strength to the coherence property. One gets a Lipschitzian complete error calculus which behaves well by images and by products and allows a quick and easy construction of the basic mathematical tools of Malliavin calculus. This allows also to revisit the delicate question of error permanency that Poincaré emphasized. This error calculus is connected with statistics by mean of the notion of Fisher information.

math.PR↗

arXiv2006

When and how an error yields a Dirichlet form

We consider a random variable $Y$ and approximations $Y\_n$, defined on the same probability space with values in the same measurable space as $Y$. We are interested in situations where the approximations $Y\_n$ allow to define a Dirichlet form in the space $L^2(P\_Y)$ where $P\_Y$ is the law of $Y$. Our approach consists in studying both biases and variances. The article attempts to propose a general theoretical framework. It is illustrated by several examples.

math.FA↗

arXiv2006

Some thoughts upon axiomatized languages with estension tools, a focus on probability theory and error calculus with Dirichlet forms

A comparison of the "theory of random sequences" developed during the twentieth century and the axiomatic approach of probability theory proposed by Kolmogorov shows the importance of sigma-additivity as extension tool. Similarly, the Cauchy criterion appears to be an extension tool for mathematical analysis. The Dirichlet forms theory possesses also such an extension tool. They are the source of the fruitfulness of these languages and the condition of their creativity. A connection is given with the so-called Richard paradox.

math.HO↗

arXiv2006

Théorème de Donsker et formes de Dirichlet

We use the language of errors to handle local Dirichlet forms with square field operator (cf [2]). Let us consider, under the hypotheses of Donsker theorem, a random walk converging weakly to a Brownian motion. If in addition the random walk is supposed to be erroneous, the convergence occurs in the sense of Dirichlet forms and induces the Ornstein-Uhlenbeck structure on the Wiener space. This quite natural result uses an extension of Donsker theorem to functions with quadratic growth. As an application we prove an invariance principle for the gradient of the maximum of the Brownian path computed by Nualart and Vives.

math.PR↗

arXiv2006

On the coarsest topology preserving continuity

We study a topology on a space of functions, called sticking topology, with the property to be the weakest among the topologies preserving continuity. In suitable frameworks, this topology preserves borelianity, local integrability, right continuity and other properties. It is coarser than the locally uniform convergence and it allows the presence of gliding humps as we show on examples. We prove relative compactness criteria for this topology and we consider some extensions.

math.GN↗