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Concept

Formule (mathématiques)

Résumé

En logique et en mathématiques, une formule est une suite finie d'objets, dotée de propriétés particulières qui rendent possible la syntaxe dans tous ces domaines. Définition Étant donné un ensemble E et une fonction de poids p: E →N, une formule est un mot extrait de E obtenu par les deux règles de construction suivantes :

un seul élément de E de poids 0 est une formule ;

si t est un élément de poids n, pour toute suite de n formules F1, F2, ...., Fn, le mot concaténé tF1F2....Fn est une formule.

On reconnaît les « mots significatifs » qui forment un sous-ensemble du monoïde libre Lo(E) construit sur E. La notation théorique introduite ici est celle dite de Łukasiewicz ou « notation polonaise » ; mais la notation communément utilisée en algèbre et en analyse est celle à parenthèses t(F2, ...., Fn) ; si t est de poids 2, on écrit (F1)t(F2) au lieu de tF1F2, et [r(F1, ...., Fm)] t [s(G1, ...., Gn)] au lieu de trF1 ....FmsG1....Gn. Étant donné une formule F, tout i

Source officielle

https://fr.wikipedia.org/wiki/Formule_(math%C3%A9matiques)

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Essays in Financial Economics

Julien Arsène Blatt

I started my PhD studies in August 2014 with a strong desire to push my own limits without knowing precisely the areas I wanted to cover in detail. To me, it was clear that I was interested by many different fields, however, I was particularly concerned with behavioural finance and with the fact that simple actions could be followed by strong market reactions. It is in this context that my supervisor, Prof. Semyon Malamud, advised me to derive/measure the consequences of the large acceptance of the RiskMetrics variance model on the price of financial assets. Indeed, this method has the advantage of providing a simple formula to estimate the volatility of any financial asset, but above all, has been used significantly by practitioners in the financial industry. The question then arises, ``Is there a link between this method and the price of financial assets?'' In order to answer this question, I have designed a simple portfolio optimization model in which agents update volatility estimates with the RiskMetrics formula. Thanks to this simple idea my first project was born and I quickly realized that I could design an elegant model. With this framework I have been able to establish the existence of a risk factor of which the economic literature was unaware. Moreover, the empirical strategy allows me to estimate the relative risk aversion coefficient independently from established procedures. Importantly, my estimates are in line with the ones obtained with these (standard) approaches. Meanwhile, I was also interested in a topic that covers a large part of all trades and is known as ``over-the-counter markets''. These markets are characterized by their high level of decentralization. Indeed, every transaction is settled directly between a buyer and a seller. In these markets, the only way to secure a trade is to find another agent that is willing to take the counter-party. I became very interested in a series of books and articles that were modeling financial assets traded under these conditions. Hence, I have started to work in this field by solving different models. I was particularly interested in understanding how the price of assets traded with this constraint would react under stressful situations, that is, when agents had to liquidate their investments. After a trial and error process, I found that my model generated puzzling results. Indeed, this model made predictions that were against traditional wisdom. Above all, that model predicted that a large level of capital mobility could impair welfare. Hence, I had eventually found the link between all the fields I wanted to cover in my thesis, where I debate the optimal allocation of capital under different types of frictions. While my first article treats the case of a centralized market with an agent who forecasts volatility using a particular method, the second article concerns how capital flows across markets when agents are subject to searching frictions. My third article is based on the second and discusses the interaction between innovation and the competition between firms supplying the same products. This article focuses on how capital is used by firms to innovate and how firms grow.

EPFL2019

Chargement

We prove the existence of an affine paving for the three-step flag Hilbert scheme that parametrizes flag of three 0-dimensional subschemes of length, respectively, n, n+1 and n+2 that are supported at the origin of the affine plane. This is done by showing that the space stratifies in smooth subvarieties, the Hilbert-Samuel's strata, each of which has an affine paving with cells of known dimension, indexed by marked Young diagrams. The affine pavings of the Hilbert-Samuel's strata allow us to prove that the Poincaré polynomials for our spaces satisfy a generating function. In the process of proving the formula for the generating function we relate combinatorially the homology of our spaces with that of known smooth subspaces of another Hilbert scheme of flags, this time of length n and n+2. As a corollary we find an affine paving and a combinatorial formula for the Poincaré of these last ambient spaces.

EPFL2016

Chargement

We study the spectrum of local operators living on a defect in a generic conformal field theory, and their coupling to the local bulk operators. We establish the existence of universal accumulation points in the spectrum at large s, s being the charge of the operators under rotations in the space transverse to the defect. Our tools include a formula that inverts the bulk to defect OPE, analogous to the Caron-Huot formula for the four-point function [1]. Analyticity of the formula in s implies that the scaling dimensions of the defect operators are aligned in Regge trajectories (Delta) over cap (s). These results require the correlator of two local operators and the defect to be bounded in a certain region, a condition that we do not prove in general. We check our conclusions against examples in perturbation theory and holography, and we make specific predictions concerning the spectrum of defect operators on Wilson lines. We also give an interpretation of the large s spectrum in the spirit of the work of Alday and Maldacena [2].

SPRINGER2018

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Essays in Financial Economics

Julien Arsène Blatt

EPFL2019