Stable distribution

In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stable if its distribution is stable. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it. Of the four parameters defining the family, most attention has been focused on the stability parameter, \alpha (see panel). Stable distributions have 0 < \alpha \leq 2, with the upper bound corresponding to the normal distribution, and \alpha=1 to the Cauchy distribution. The distributions have undefined variance for \alpha < 2, and undefined mean for \alpha \leq 1. The importance of stable probability distributions is that they are "attractors" for properly normed sums of independent and i

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The course will provide a synopsis of the chemistry of f elements (lanthanides and actinides) covering structure, bonding, redox and spectroscopic properties and reactivity. The coordination and organometallic chemistry of these ions will be discussed with an overview of their main applications.

Ce cours introduit les systèmes dynamiques pour modéliser des réseaux biologiques simples. L'analyse qualitative de modèles dynamiques non-linéaires est développée de pair avec des simulations numériques. L'accent est mis sur les exemples biologiques.

This course discusses the interaction between atoms and visible electro-magnetic radiation and introduces the main instrumentation for light detection and spectroscopy. The principles of LASER light sources are described, providing notions of nonlinear and ultrafast optics.

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Normal distribution

In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function

Cauchy distribution

The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Ca

Heavy-tailed distribution

In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. In many

Les mathématiques financières robustes, le maritime et les FFA

Jérôme Reboulleau

Powerful mathematical tools have been developed for trading in stocks and bonds, but other markets that are equally important for the globalized world have to some extent been neglected. We decided to study the shipping market as an new area of development in mathematical finance. The market in shipping derivatives (FFA and FOSVA) has only been developed after 2000 and now exhibits impressive growth. Financial actors have entered the field, but it is still largely undiscovered by institutional investors. The first part of the work was to identify the characteristics of the market in shipping, i.e. the segmentation and the volatility. Because the shipping business is old-fashioned, even the leading actors on the world stage (ship owners and banks) are using macro-economic models to forecast the rates. If the macro-economic models are logical and make sense, they fail to predict. For example, the factor port congestion has been much cited during the last few years, but it is clearly very difficult to control and is simply an indicator of traffic. From our own experience it appears that most ship owners are in fact market driven and rather bad at anticipating trends. Due to their ability to capture large moves, we chose to consider Lévy processes for the underlying price process. Compared with the macro-economic approach, the main advantage is the uniform and systematic structure this imposed on the models. We get in each case a favorable result for our technology and a gain in forecasting accuracy of around 10% depending on the maturity. The global distribution is more effectively modelled and the tails of the distribution are particularly well represented. This model can be used to forecast the market but also to evaluate the risk, for example, by computing the VaR. An important limitation is the non-robustness in the estimation of the Lévy processes. The use of robust estimators reinforces the information obtained from the observed data. Because maximum likelihood estimation is not easy to compute with complex processes, we only consider some very general robust score functions to manage the technical problems. Two new class of robust estimators are suggested. These are based on the work of F. Hampel ([29]) and P. Huber ([30]) using influence functions. The main idea is to bound the maximum likelihood score function. By doing this a bias is created in the parameters estimation, which can be corrected by using a modification of the following type and as proposed by F. Hampel. The procedure for finding a robust estimating equation is thus decomposed into two consecutive steps : Subtract the bias correction and then Bound the score function. In the case of complex Lévy processes, the bias correction is difficult to compute and generally unknown. We have developed a pragmatic solution by inverting the Hampel's procedure. Bound the score function and then Correct for the bias. The price is a loss of the theoretical properties of our estimators, besides the procedure converges to maximum likelihood estimate. A second solution to for achieving robust estimation is presented. It considers the limiting case when the upper and lower bounds tend to zero and leads to B-robust estimators. Because of the complexity of the Lévy distributions, this leads to identification problems.

EPFL2008

Prédiction spatiale en présence d'erreurs substitutives et modélisation par drap stable

Baptiste Fournier

In geostatistics, the presence of outlying data is more the rule than the exception. Moreover, the statistical analysis of data contaminated by outliers requires caution, particularly when a spatial dependence exists. In order to take into account these possible outliers during the adjustment of the spatial process, a new modeling tool, called the substitutive errors model, is proposed. The optimal prediction in the least squares sense is derived and its properties are studied. Because of its complexity, this estimator needs in practice to be numerically approximated. An automated algorithm is proposed in this thesis. This method is based on an ordering of the observations with respect to the specified spatial process of interest, with the values least in agreement being included towards the end of the ordering. It proves to be useful in case of masked multiple outliers or nonstationary clusters. Simulations are carried out to illustrate its performances and to compare it to other forecasts, robust or not. An application to real data is provided as an illustration of its practical usefulness. The second part of this work also deals with the presence of spatial heterogeneity. One could say that the proposed model offers a characterization of this heterogeneity rather than estimating the locations and sizes of outliers. It is based on the theory of bidimensional α-stable motion. This represents a generalization of the unidimensional Brownian motion. In particular, the stability parameter α can be seen as a measure of the distance between the observations and the hypothesis of a Gaussian distribution. A method of estimation for the parameters of such a process is presented, based on a numerical constrained optimization of the likelihood. Its performances are illustrated by means of simulations. An application ends this second part.

EPFL2007

Stochastic partial differential equations driven by Lévy white noises

Thomas Marie Jean-Baptiste Humeau

We study various aspects of stochastic partial differential equations driven by Lévy white noise. This driving noise, which is a generalization of Gaussian white noise, can be viewed either as a generalized random process or as an independently scattered random measure. After unifying these approaches and establishing appropriate stochastic integral representations, we show that a necessary and sufficient condition for a Lévy white noise to have values in the space of tempered Schwartz distributions, is that the underlying Lévy measure have a positive absolute moment. In the case of a linear stochastic partial differential equation with a general differential operator and driven by a symmetric pure jump Lévy white noise, we show that when the mild solution is locally Lebesgue integrable, then it is equal to the generalized solution, and that a random field representation exists for the generalized solution if and only if the fundamental solution of the operator has certain integrability properties. In that case, we show that the random field representation is equal to the mild solution. For this purpose, a new stochastic Fubini theorem is proved. These results are applied to the linear stochastic heat and wave equations driven by a symmetric alpha-stable noise. We then study the non-linear stochastic heat equation driven by a general type of Lévy white noise, possibly with heavy tails and non-summable small jumps. Our framework includes in particular the alpha-stable noise. In the case of the equation on the whole space, we show that the law of the solution that we construct does not depend on the space variable. Then we show in various domains D that the solution u to the stochastic heat equation is such that t -> u(t,·) has a càdlàg version in a fractional Sobolev space of order r < -d /2. Finally, we show that the partial functions have a continuous version under some optimal moment conditions. In the alpha-stable case, we show that for the choices of alpha for which this moment condition is not satisfied, the sample paths of the partial functions are unbounded on any non-empty open subset.

EPFL2017