Hopf hypersurfaces in pseudo-Riemannian complex and para-complex space forms

The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic hypersurfaces, nor real hypersurfaces with parallel shape operator in such spaces. Denoting by J be the complex or para-complex structure of a pseudo-complex or para-complex space form respectively, a non-degenerate hypersurface of such space with unit normal vector field N is said to be Hopf if the tangent vector field JN is a principal direction. It is proved that if a hypersurface is Hopf, then the corresponding principal curvature (the Hopf curvature) is constant. It is also observed that in some cases a Hopf hypersurface must be, locally, a tube over a complex (or para-complex) submanifold, thus generalizing previous results of Cecil, Ryan and Montiel.

A representação de uma árvore para uma nova forma de ver o tempo e o espaço : desenho A 12º ano

Relatório da prática de ensino supervisionada, Mestrado em Ensino de Artes Visuais, Universidade de Lisboa, 2014

Traités scientifiques d'Isaac MARTIN, secrétaire du roi.

Contient : I Traités Originaux portant le nom de l'auteur ; « Traicté de la Perspective », avec de nombreuses figures ; à la suite (pages 89 et suiv.), devait se trouver une série de dessins à la plume, dont il ne reste plus que deux (page 89, et page 91, anciennement cotée 103) ; « Arithmétique » ; « Practique de géométrie » ; II Traités Originaux sans nom d'auteur, mais d'Isaac Martin ; « Des règles de la Proportion » ; « De la Progression » arithmétique ; « De la Progression géométrique » ; « Des extractions de Racine » ; « Questions militaires sur la Racine quarrée » ; Figures de géométrie

Latent Variable Models for Stochastic Discount Factors.

Latent variable models in finance originate both from asset pricing theory and time series analysis. These two strands of literature appeal to two different concepts of latent structures, which are both useful to reduce the dimension of a statistical model specified for a multivariate time series of asset prices. In the CAPM or APT beta pricing models, the dimension reduction is cross-sectional in nature, while in time-series state-space models, dimension is reduced longitudinally by assuming conditional independence between consecutive returns, given a small number of state variables. In this paper, we use the concept of Stochastic Discount Factor (SDF) or pricing kernel as a unifying principle to integrate these two concepts of latent variables. Beta pricing relations amount to characterize the factors as a basis of a vectorial space for the SDF. The coefficients of the SDF with respect to the factors are specified as deterministic functions of some state variables which summarize their dynamics. In beta pricing models, it is often said that only the factorial risk is compensated since the remaining idiosyncratic risk is diversifiable. Implicitly, this argument can be interpreted as a conditional cross-sectional factor structure, that is, a conditional independence between contemporaneous returns of a large number of assets, given a small number of factors, like in standard Factor Analysis. We provide this unifying analysis in the context of conditional equilibrium beta pricing as well as asset pricing with stochastic volatility, stochastic interest rates and other state variables. We address the general issue of econometric specifications of dynamic asset pricing models, which cover the modern literature on conditionally heteroskedastic factor models as well as equilibrium-based asset pricing models with an intertemporal specification of preferences and market fundamentals. We interpret various instantaneous causality relationships between state variables and market fundamentals as leverage effects and discuss their central role relative to the validity of standard CAPM-like stock pricing and preference-free option pricing.

Asymmetric Smiles, Leverage Effects and Structural Parameters.

In this paper, we characterize the asymmetries of the smile through multiple leverage effects in a stochastic dynamic asset pricing framework. The dependence between price movements and future volatility is introduced through a set of latent state variables. These latent variables can capture not only the volatility risk and the interest rate risk which potentially affect option prices, but also any kind of correlation risk and jump risk. The standard financial leverage effect is produced by a cross-correlation effect between the state variables which enter into the stochastic volatility process of the stock price and the stock price process itself. However, we provide a more general framework where asymmetric implied volatility curves result from any source of instantaneous correlation between the state variables and either the return on the stock or the stochastic discount factor. In order to draw the shapes of the implied volatility curves generated by a model with latent variables, we specify an equilibrium-based stochastic discount factor with time non-separable preferences. When we calibrate this model to empirically reasonable values of the parameters, we are able to reproduce the various types of implied volatility curves inferred from option market data.

Empirical Assessment of an Intertemporal Option Pricing Model with Latent variables

This paper assesses the empirical performance of an intertemporal option pricing model with latent variables which generalizes the Hull-White stochastic volatility formula. Using this generalized formula in an ad-hoc fashion to extract two implicit parameters and forecast next day S&P 500 option prices, we obtain similar pricing errors than with implied volatility alone as in the Hull-White case. When we specialize this model to an equilibrium recursive utility model, we show through simulations that option prices are more informative than stock prices about the structural parameters of the model. We also show that a simple method of moments with a panel of option prices provides good estimates of the parameters of the model. This lays the ground for an empirical assessment of this equilibrium model with S&P 500 option prices in terms of pricing errors.

A Theoretical Comparison Between Integrated and Realized Volatilies

In this paper, we provide both qualitative and quantitative measures of the cost of measuring the integrated volatility by the realized volatility when the frequency of observation is fixed. We start by characterizing for a general diffusion the difference between the realized and the integrated volatilities for a given frequency of observations. Then, we compute the mean and variance of this noise and the correlation between the noise and the integrated volatility in the Eigenfunction Stochastic Volatility model of Meddahi (2001a). This model has, as special examples, log-normal, affine, and GARCH diffusion models. Using some previous empirical works, we show that the standard deviation of the noise is not negligible with respect to the mean and the standard deviation of the integrated volatility, even if one considers returns at five minutes. We also propose a simple approach to capture the information about the integrated volatility contained in the returns through the leverage effect.

An Eigenfunction Approach for Volatility Modeling.

In this paper, we introduce a new approach for volatility modeling in discrete and continuous time. We follow the stochastic volatility literature by assuming that the variance is a function of a state variable. However, instead of assuming that the loading function is ad hoc (e.g., exponential or affine), we assume that it is a linear combination of the eigenfunctions of the conditional expectation (resp. infinitesimal generator) operator associated to the state variable in discrete (resp. continuous) time. Special examples are the popular log-normal and square-root models where the eigenfunctions are the Hermite and Laguerre polynomials respectively. The eigenfunction approach has at least six advantages: i) it is general since any square integrable function may be written as a linear combination of the eigenfunctions; ii) the orthogonality of the eigenfunctions leads to the traditional interpretations of the linear principal components analysis; iii) the implied dynamics of the variance and squared return processes are ARMA and, hence, simple for forecasting and inference purposes; (iv) more importantly, this generates fat tails for the variance and returns processes; v) in contrast to popular models, the variance of the variance is a flexible function of the variance; vi) these models are closed under temporal aggregation.

Termes de l'échange endogènes et cycles économiques réels : une application à la Côte-d'Ivoire

À l’aide d’un modèle de cycles réels, la présente étude vise à expliquer, de façon endogène, les fluctuations des termes de l’échange en Côte-d’Ivoire. Pour ce faire, nous cherchons principalement à répondre aux deux questions suivantes : les chocs d’offre et de demande sur le marché d’exportation suffisent-ils à expliquer les variations des termes de l’échange? Et quelle est leur importance relative dans la dynamique des termes de l’échange? Les résultats montrent que les deux chocs considérés expliquent bien la volatilité des termes de l’échange. Nous avons noté que ces deux sources d’impulsions ont un impact significatif sur les fluctuations économiques en Côte-d’Ivoire.

Risk Aversion, Intertemporal Substitution, and Option Pricing

This paper develops a general stochastic framework and an equilibrium asset pricing model that make clear how attitudes towards intertemporal substitution and risk matter for option pricing. In particular, we show under which statistical conditions option pricing formulas are not preference-free, in other words, when preferences are not hidden in the stock and bond prices as they are in the standard Black and Scholes (BS) or Hull and White (HW) pricing formulas. The dependence of option prices on preference parameters comes from several instantaneous causality effects such as the so-called leverage effect. We also emphasize that the most standard asset pricing models (CAPM for the stock and BS or HW preference-free option pricing) are valid under the same stochastic setting (typically the absence of leverage effect), regardless of preference parameter values. Even though we propose a general non-preference-free option pricing formula, we always keep in mind that the BS formula is dominant both as a theoretical reference model and as a tool for practitioners. Another contribution of the paper is to characterize why the BS formula is such a benchmark. We show that, as soon as we are ready to accept a basic property of option prices, namely their homogeneity of degree one with respect to the pair formed by the underlying stock price and the strike price, the necessary statistical hypotheses for homogeneity provide BS-shaped option prices in equilibrium. This BS-shaped option-pricing formula allows us to derive interesting characterizations of the volatility smile, that is, the pattern of BS implicit volatilities as a function of the option moneyness. First, the asymmetry of the smile is shown to be equivalent to a particular form of asymmetry of the equivalent martingale measure. Second, this asymmetry appears precisely when there is either a premium on an instantaneous interest rate risk or on a generalized leverage effect or both, in other words, whenever the option pricing formula is not preference-free. Therefore, the main conclusion of our analysis for practitioners should be that an asymmetric smile is indicative of the relevance of preference parameters to price options.

Asymptotic Approximations in the Near-Integrated Model with a Non-Zero Initial Condition

This paper considers various asymptotic approximations in the near-integrated firstorder autoregressive model with a non-zero initial condition. We first extend the work of Knight and Satchell (1993), who considered the random walk case with a zero initial condition, to derive the expansion of the relevant joint moment generating function in this more general framework. We also consider, as alternative approximations, the stochastic expansion of Phillips (1987c) and the continuous time approximation of Perron (1991). We assess how these alternative methods provide or not an adequate approximation to the finite-sample distribution of the least-squares estimator in a first-order autoregressive model. The results show that, when the initial condition is non-zero, Perron's (1991) continuous time approximation performs very well while the others only offer improvements when the initial condition is zero.

Aggregations and Marginalization of GARCH and Stochastic Volatility Models

The GARCH and Stochastic Volatility paradigms are often brought into conflict as two competitive views of the appropriate conditional variance concept : conditional variance given past values of the same series or conditional variance given a larger past information (including possibly unobservable state variables). The main thesis of this paper is that, since in general the econometrician has no idea about something like a structural level of disaggregation, a well-written volatility model should be specified in such a way that one is always allowed to reduce the information set without invalidating the model. To this respect, the debate between observable past information (in the GARCH spirit) versus unobservable conditioning information (in the state-space spirit) is irrelevant. In this paper, we stress a square-root autoregressive stochastic volatility (SR-SARV) model which remains true to the GARCH paradigm of ARMA dynamics for squared innovations but weakens the GARCH structure in order to obtain required robustness properties with respect to various kinds of aggregation. It is shown that the lack of robustness of the usual GARCH setting is due to two very restrictive assumptions : perfect linear correlation between squared innovations and conditional variance on the one hand and linear relationship between the conditional variance of the future conditional variance and the squared conditional variance on the other hand. By relaxing these assumptions, thanks to a state-space setting, we obtain aggregation results without renouncing to the conditional variance concept (and related leverage effects), as it is the case for the recently suggested weak GARCH model which gets aggregation results by replacing conditional expectations by linear projections on symmetric past innovations. Moreover, unlike the weak GARCH literature, we are able to define multivariate models, including higher order dynamics and risk premiums (in the spirit of GARCH (p,p) and GARCH in mean) and to derive conditional moment restrictions well suited for statistical inference. Finally, we are able to characterize the exact relationships between our SR-SARV models (including higher order dynamics, leverage effect and in-mean effect), usual GARCH models and continuous time stochastic volatility models, so that previous results about aggregation of weak GARCH and continuous time GARCH modeling can be recovered in our framework.

Methods to Estimate Dynamic Stochastic General Equilibrium Models

This paper employs the one-sector Real Business Cycle model as a testing ground for four different procedures to estimate Dynamic Stochastic General Equilibrium (DSGE) models. The procedures are: 1 ) Maximum Likelihood, with and without measurement errors and incorporating Bayesian priors, 2) Generalized Method of Moments, 3) Simulated Method of Moments, and 4) Indirect Inference. Monte Carlo analysis indicates that all procedures deliver reasonably good estimates under the null hypothesis. However, there are substantial differences in statistical and computational efficiency in the small samples currently available to estimate DSGE models. GMM and SMM appear to be more robust to misspecification than the alternative procedures. The implications of the stochastic singularity of DSGE models for each estimation method are fully discussed.

A Theory of Random Consumer Demand

This paper presents a new theory of random consumer demand. The primitive is a collection of probability distributions, rather than a binary preference. Various assumptions constrain these distributions, including analogues of common assumptions about preferences such as transitivity, monotonicity and convexity. Two results establish a complete representation of theoretically consistent random demand. The purpose of this theory of random consumer demand is application to empirical consumer demand problems. To this end, the theory has several desirable properties. It is intrinsically stochastic, so the econometrician can apply it directly without adding extrinsic randomness in the form of residuals. Random demand is parsimoniously represented by a single function on the consumption set. Finally, we have a practical method for statistical inference based on the theory, described in McCausland (2004), a companion paper.

Anatomo-functional magnetic resonance imaging of the spinal cord and its application to the characterization of spinal lesions in cats

Les lésions de la moelle épinière ont un impact significatif sur la qualité de la vie car elles peuvent induire des déficits moteurs (paralysie) et sensoriels. Ces déficits évoluent dans le temps à mesure que le système nerveux central se réorganise, en impliquant des mécanismes physiologiques et neurochimiques encore mal connus. L'ampleur de ces déficits ainsi que le processus de réhabilitation dépendent fortement des voies anatomiques qui ont été altérées dans la moelle épinière. Il est donc crucial de pouvoir attester l'intégrité de la matière blanche après une lésion spinale et évaluer quantitativement l'état fonctionnel des neurones spinaux. Un grand intérêt de l'imagerie par résonance magnétique (IRM) est qu'elle permet d'imager de façon non invasive les propriétés fonctionnelles et anatomiques du système nerveux central. Le premier objectif de ce projet de thèse a été de développer l'IRM de diffusion afin d'évaluer l'intégrité des axones de la matière blanche après une lésion médullaire. Le deuxième objectif a été d'évaluer dans quelle mesure l'IRM fonctionnelle permet de mesurer l'activité des neurones de la moelle épinière. Bien que largement appliquées au cerveau, l'IRM de diffusion et l'IRM fonctionnelle de la moelle épinière sont plus problématiques. Les difficultés associées à l'IRM de la moelle épinière relèvent de sa fine géométrie (environ 1 cm de diamètre chez l'humain), de la présence de mouvements d'origine physiologique (cardiaques et respiratoires) et de la présence d'artefacts de susceptibilité magnétique induits par les inhomogénéités de champ, notamment au niveau des disques intervertébraux et des poumons. L'objectif principal de cette thèse a donc été de développer des méthodes permettant de contourner ces difficultés. Ce développement a notamment reposé sur l'optimisation des paramètres d'acquisition d'images anatomiques, d'images pondérées en diffusion et de données fonctionnelles chez le chat et chez l'humain sur un IRM à 3 Tesla. En outre, diverses stratégies ont été étudiées afin de corriger les distorsions d'images induites par les artefacts de susceptibilité magnétique, et une étude a été menée sur la sensibilité et la spécificité de l'IRM fonctionnelle de la moelle épinière. Les résultats de ces études démontrent la faisabilité d'acquérir des images pondérées en diffusion de haute qualité, et d'évaluer l'intégrité de voies spinales spécifiques après lésion complète et partielle. De plus, l'activité des neurones spinaux a pu être détectée par IRM fonctionnelle chez des chats anesthésiés. Bien qu'encourageants, ces résultats mettent en lumière la nécessité de développer davantage ces nouvelles techniques. L'existence d'un outil de neuroimagerie fiable et robuste, capable de confirmer les paramètres cliniques, permettrait d'améliorer le diagnostic et le pronostic chez les patients atteints de lésions médullaires. Un des enjeux majeurs serait de suivre et de valider l'effet de diverses stratégies thérapeutiques. De telles outils représentent un espoir immense pour nombre de personnes souffrant de traumatismes et de maladies neurodégénératives telles que les lésions de la moelle épinière, les tumeurs spinales, la sclérose en plaques et la sclérose latérale amyotrophique.