Applications of stochastic analysis in finance and statistical inference

First: A continuous-time version of Kyle's model (Kyle 1985), known as the Back's model (Back 1992), of asset pricing with asymmetric information, is studied. A larger class of price processes and of noise traders' processes are studied. The price process, as in Kyle's model, is allowed to depend on the path of the market order. The process of the noise traders' is an inhomogeneous Lévy process. Solutions are found by the Hamilton-Jacobi-Bellman equations. With the insider being risk-neutral, the price pressure is constant, and there is no equilibirium in the presence of jumps. If the insider is risk-averse, there is no equilibirium in the presence of either jumps or drifts. Also, it is analised when the release time is unknown. A general relation is established between the problem of finding an equilibrium and of enlargement of filtrations. Random announcement time is random is also considered. In such a case the market is not fully efficient and there exists equilibrium if the sensitivity of prices with respect to the global demand is time decreasing according with the distribution of the random time. Second: Power variations. it is considered, the asymptotic behavior of the power variation of processes of the form _integral_0^t u(s-)dS(s), where S_ is an alpha-stable process with index of stability 0&alpha&2 and the integral is an Itô integral. Stable convergence of corresponding fluctuations is established. These results provide statistical tools to infer the process u from discrete observations. Third: A bond market is studied where short rates r(t) evolve as an integral of g(t-s)sigma(s) with respect to W(ds), where g and sigma are deterministic and W is the stochastic Wiener measure. Processes of this type are particular cases of ambit processes. These processes are in general not of the semimartingale kind.

Statistical analysis and modelling of weather radar beam propagation conditions in the Po Valley (Italy)

Ground clutter caused by anomalous propagation (anaprop) can affect seriously radar rain rate estimates, particularly in fully automatic radar processing systems, and, if not filtered, can produce frequent false alarms. A statistical study of anomalous propagation detected from two operational C-band radars in the northern Italian region of Emilia Romagna is discussed, paying particular attention to its diurnal and seasonal variability. The analysis shows a high incidence of anaprop in summer, mainly in the morning and evening, due to the humid and hot summer climate of the Po Valley, particularly in the coastal zone. Thereafter, a comparison between different techniques and datasets to retrieve the vertical profile of the refractive index gradient in the boundary layer is also presented. In particular, their capability to detect anomalous propagation conditions is compared. Furthermore, beam path trajectories are simulated using a multilayer ray-tracing model and the influence of the propagation conditions on the beam trajectory and shape is examined. High resolution radiosounding data are identified as the best available dataset to reproduce accurately the local propagation conditions, while lower resolution standard TEMP data suffers from interpolation degradation and Numerical Weather Prediction model data (Lokal Model) are able to retrieve a tendency to superrefraction but not to detect ducting conditions. Observing the ray tracing of the centre, lower and upper limits of the radar antenna 3-dB half-power main beam lobe it is concluded that ducting layers produce a change in the measured volume and in the power distribution that can lead to an additional error in the reflectivity estimate and, subsequently, in the estimated rainfall rate.

The generalized index of maximum and minimum level and its application in decision making

[spa] El índice del máximo y el mínimo nivel es una técnica muy útil, especialmente para toma de decisiones, que usa la distancia de Hamming y el coeficiente de adecuación en el mismo problema. En este trabajo, se propone una generalización a través de utilizar medias generalizadas y cuasi aritméticas. A estos operadores de agregación, se les denominará el índice del máximo y el mínimo nivel medio ponderado ordenado generalizado (GOWAIMAM) y cuasi aritmético (Quasi-OWAIMAM). Estos nuevos operadores generalizan una amplia gama de casos particulares como el índice del máximo y el mínimo nivel generalizado (GIMAM), el OWAIMAM, y otros. También se desarrolla una aplicación en la toma de decisiones sobre selección de productos.

The generalized index of maximum and minimum level and its application in decision making

[spa] El índice del máximo y el mínimo nivel es una técnica muy útil, especialmente para toma de decisiones, que usa la distancia de Hamming y el coeficiente de adecuación en el mismo problema. En este trabajo, se propone una generalización a través de utilizar medias generalizadas y cuasi aritméticas. A estos operadores de agregación, se les denominará el índice del máximo y el mínimo nivel medio ponderado ordenado generalizado (GOWAIMAM) y cuasi aritmético (Quasi-OWAIMAM). Estos nuevos operadores generalizan una amplia gama de casos particulares como el índice del máximo y el mínimo nivel generalizado (GIMAM), el OWAIMAM, y otros. También se desarrolla una aplicación en la toma de decisiones sobre selección de productos.

The statistical accuracy of surveys on business and economic perspectives: a case study

In this paper, we develop a new decision making model and apply it in political Surveys of economic climate collect opinions of managers about the short-term future evolution of their business. Interviews are carried out on a regular basis and responses measure optimistic, neutral or pessimistic views about the economic perspectives. We propose a method to evaluate the sampling error of the average opinion derived from a particular type of survey data. Our variance estimate is useful to interpret historical trends and to decide whether changes in the index from one period to another are due to a structural change or whether ups and downs can be attributed to sampling randomness. An illustration using real data from a survey of business managers opinions is discussed.

The Total Deviation Index estimated by Tolerance Intervals to evaluate the concordance of measurement devices

Background In an agreement assay, it is of interest to evaluate the degree of agreement between the different methods (devices, instruments or observers) used to measure the same characteristic. We propose in this study a technical simplification for inference about the total deviation index (TDI) estimate to assess agreement between two devices of normally-distributed measurements and describe its utility to evaluate inter- and intra-rater agreement if more than one reading per subject is available for each device. Methods We propose to estimate the TDI by constructing a probability interval of the difference in paired measurements between devices, and thereafter, we derive a tolerance interval (TI) procedure as a natural way to make inferences about probability limit estimates. We also describe how the proposed method can be used to compute bounds of the coverage probability. Results The approach is illustrated in a real case example where the agreement between two instruments, a handle mercury sphygmomanometer device and an OMRON 711 automatic device, is assessed in a sample of 384 subjects where measures of systolic blood pressure were taken twice by each device. A simulation study procedure is implemented to evaluate and compare the accuracy of the approach to two already established methods, showing that the TI approximation produces accurate empirical confidence levels which are reasonably close to the nominal confidence level. Conclusions The method proposed is straightforward since the TDI estimate is derived directly from a probability interval of a normally-distributed variable in its original scale, without further transformations. Thereafter, a natural way of making inferences about this estimate is to derive the appropriate TI. Constructions of TI based on normal populations are implemented in most standard statistical packages, thus making it simpler for any practitioner to implement our proposal to assess agreement.

Measuring Intermediary Determinants of Early Childhood Health: A Composite Index Comparing Colombian Departments

In recent years there has been growing interest in composite indicators as an efficient tool of analysis and a method of prioritizing policies. This paper presents a composite index of intermediary determinants of child health using a multivariate statistical approach. The index shows how specific determinants of child health vary across Colombian departments (administrative subdivisions). We used data collected from the 2010 Colombian Demographic and Health Survey (DHS) for 32 departments and the capital city, Bogotá. Adapting the conceptual framework of Commission on Social Determinants of Health (CSDH), five dimensions related to child health are represented in the index: material circumstances, behavioural factors, psychosocial factors, biological factors and the health system. In order to generate the weight of the variables, and taking into account the discrete nature of the data, principal component analysis (PCA) using polychoric correlations was employed in constructing the index. From this method five principal components were selected. The index was estimated using a weighted average of the retained components. A hierarchical cluster analysis was also carried out. The results show that the biggest differences in intermediary determinants of child health are associated with health care before and during delivery.

Measuring early childhood health : a composite index comparing Colombian departments

This paper presents a composite index of early childhood health using a multivariate statistical approach. The index shows how child health varies across Colombian departments, -administrative subdivisions-. In recent years there has been growing interest in composite indicators as an efficient analysis tool and a way of prioritizing policies. These indicators not only enable multi-dimensional phenomena to be simplified but also make it easier to measure, visualize, monitor and compare a country’s performance in particular issues. We used data collected from the Colombian Demographic and Health Survey, DHS, for 32 departments and the capital city, Bogotá, in 2005 and 2010. The variables included in the index provide a measure of three dimensions related to child health: health status, health determinants and the health system. In order to generate the weight of the variables and take into account the discrete nature of the data, we employed a principal component analysis, PCA, using polychoric correlation. From this method, five principal components were selected. The index was estimated using a weighted average of the components retained. A hierarchical cluster analysis was also carried out. We observed that the departments ranking in the lowest positions are located on the Colombian periphery. They are departments with low per capita incomes and they present critical social indicators. The results suggest that the regional disparities in child health may be associated with differences in parental characteristics, household conditions and economic development levels, which makes clear the importance of context in the study of child health in Colombia.

A Theta-Band EEG Based Index for Early Diagnosis of Alzheimer’s Disease

Despite recent advances, early diagnosis of Alzheimer’s disease (AD) from electroencephalography (EEG) remains a difficult task. In this paper, we offer an added measure through which such early diagnoses can potentially be improved. One feature that has been used for discriminative classification is changes in EEG synchrony. So far, only the decrease of synchrony in the higher frequencies has been deeply analyzed. In this paper, we investigate the increase of synchrony found in narrow frequency ranges within the θ band. This particular increase of synchrony is used with the well-known decrease of synchrony in the band to enhance detectable differences between AD patients and healthy subjects. We propose a new synchrony ratio that maximizes the differences between two populations. The ratio is tested using two different data sets, one of them containing mild cognitive impairment patients and healthy subjects, and another one, containing mild AD patients and healthy subjects. The results presented in this paper show that classification rate is improved, and the statistical difference between AD patients and healthy subjects is increased using the proposed ratio.

Infinite index subgroups and finiteness properties of intersections of geometrically finite groups

We explore which types of finiteness properties are possible for intersections of geometrically finite groups of isometries in negatively curved symmetric rank one spaces. Our main tool is a twist construction which takes as input a geometrically finite group containing a normal subgroup of infinite index with given finiteness properties and infinite Abelian quotient, and produces a pair of geometrically finite groups whose intersection is isomorphic to the normal subgroup.

Subgroups of small index in Aut(Fn) and Kazhdan's property (T)

We give a series of interesting subgroups of finite index in Aut(Fn). One of them has index 42 in Aut(F3) and infinite abelianization. This implies that Aut(F3) does not have Kazhdan’s property (T) (see [3] and [6] for another proofs). We proved also that every subgroup of finite index in Aut(Fn), n &= 3, which contains the subgroup of IA-automorphisms, has a finite abelianization.

Augment de la resolució en la reconstrucció de les temperatures de l'aigua i de l'aire del llac Baikal (Sibèria) dels últims 60.000 anys

Aquest estudi consisteix en l’augment de la resolució en la reconstrucció de la temperatura de l’aigua i l’aire del llac Baikal durant els últims 60.000 anys mitjançant l’ús de les proxies de reconstrucció de la temperatura TEX86 i MAAT, i la d’aportació de matèria orgànica d’origen terrestre, el BIT. L’objectiu general d’aquesta investigació és incrementar la resolució temporal en el mostreig del testimoni CON-01-603-02 per tal de millorar el registre de dades obtingudes i d’aquesta manera poder contrastar la hipòtesi de la interconnexió climàtica global, així com identificar esdeveniments climàtics sobtats, tals com els Heinrich events i els esdeveniments D-O. Un cop obtinguts els resultats s’ha realitzat l’anàlisi de la qualitat i fiabilitat de les dades a les resolucions de 5, 10 i 20 Kanys, i es conclou que existeixen diferències globals estadísticament significatives amb els resultats realitzats per Escala et al., (r.n.p [resultats no publicats]), la resolució dels quals es volia augmentar. S’han tractat d’anular aquestes diferències restant o bé sumant la diferència mitjana obtinguda entre les dues mostres en cada un dels intervals de 5 Kanys en què s’han donat aquestes. Els resultats integrats d’Escala et al.,(r.n.p) i els d’aquest estudi, aporten dades que recolzen l’hipòtesi de la interconnexió climàtica global, ja que al comparar-los amb els registres climàtics de Grenlàndia (GRIP2) i l’Antàrtica (Vostok) mostren respostes similars tant per les forces de Milankovitch com per les de subMilankovitch.

Calculation of the variance in surveys of the economic climate

Public opinion surveys have become progressively incorporated into systems of official statistics. Surveys of the economic climate are usually qualitative because they collect opinions of businesspeople and/or experts about the long-term indicators described by a number of variables. In such cases the responses are expressed in ordinal numbers, that is, the respondents verbally report, for example, whether during a given trimester the sales or the new orders have increased, decreased or remained the same as in the previous trimester. These data allow to calculate the percent of respondents in the total population (results are extrapolated), who select every one of the three options. Data are often presented in the form of an index calculated as the difference between the percent of those who claim that a given variable has improved in value and of those who claim that it has deteriorated. As in any survey conducted on a sample the question of the measurement of the sample error of the results has to be addressed, since the error influences both the reliability of the results and the calculation of the sample size adequate for a desired confidence interval. The results presented here are based on data from the Survey of the Business Climate (Encuesta de Clima Empresarial) developed through the collaboration of the Statistical Institute of Catalonia (Institut d’Estadística de Catalunya) with the Chambers of Commerce (Cámaras de Comercio) of Sabadell and Terrassa.

Statistical properties of subgroups of free groups

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so-called word-based distribution: subgroups are generated (finite presentations are determined) by randomly chosen k-tuples of reduced words, whose maximal length is allowed to tend to infinity. In this paper we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the so-called graph-based distribution, recently introduced by Bassino, Nicaud and Weil. Here, subgroups (and finite presentations) are determined by randomly chosen Stallings graphs whose number of vertices tends to infinity. Our results show that these two distributions behave quite differently from each other, shedding a new light on which properties of finitely generated subgroups can be considered frequent or rare. For example, we show that malnormal subgroups of a free group are negligible in the raph-based distribution, while they are exponentially generic in the word-based distribution. Quite surprisingly, a random finite presentation generically presents the trivial group in this new distribution, while in the classical one it is known to generically present an infinite hyperbolic group.

The eta invariant and equivariant index of transversally elliptic operators

We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a G-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications, we obtain an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years.