Free inertial processes driven by Gaussian noise: Probability distributions, anomalous diffusion and fractal behavior

We study the motion of an unbound particle under the influence of a random force modeled as Gaussian colored noise with an arbitrary correlation function. We derive exact equations for the joint and marginal probability density functions and find the associated solutions. We analyze in detail anomalous diffusion behaviors along with the fractal structure of the trajectories of the particle and explore possible connections between dynamical exponents of the variance and the fractal dimension of the trajectories.

Extreme times for volatility processes

Extreme times techniques, generally applied to nonequilibrium statistical mechanical processes, are also useful for a better understanding of financial markets. We present a detailed study on the mean first-passage time for the volatility of return time series. The empirical results extracted from daily data of major indices seem to follow the same law regardless of the kind of index thus suggesting an universal pattern. The empirical mean first-passage time to a certain level L is fairly different from that of the Wiener process showing a dissimilar behavior depending on whether L is higher or lower than the average volatility. All of this indicates a more complex dynamics in which a reverting force drives volatility toward its mean value. We thus present the mean first-passage time expressions of the most common stochastic volatility models whose approach is comparable to the random diffusion description. We discuss asymptotic approximations of these models and confront them to empirical results with a good agreement with the exponential Ornstein-Uhlenbeck model.

SIMSAFADIM-CLASTIC: A new approach to mathematical 3D forward simulation modelling for terrigenous and carbonate marine sedimentation

Most sedimentary modelling programs developed in recent years focus on either terrigenous or carbonate marine sedimentation. Nevertheless, only a few programs have attempted to consider mixed terrigenous-carbonate sedimentation, and most of these are two-dimensional, which is a major restriction since geological processes take place in 3D. This paper presents the basic concepts of a new 3D mathematical forward simulation model for clastic sediments, which was developed from SIMSAFADIM, a previous 3D carbonate sedimentation model. The new extended model, SIMSAFADIM-CLASTIC, simulates processes of autochthonous marine carbonate production and accumulation, together with clastic transport and sedimentation in three dimensions of both carbonate and terrigenous sediments. Other models and modelling strategies may also provide realistic and efficient tools for prediction of stratigraphic architecture and facies distribution of sedimentary deposits. However, SIMSAFADIM-CLASTIC becomes an innovative model that attempts to simulate different sediment types using a process-based approach, therefore being a useful tool for 3D prediction of stratigraphic architecture and facies distribution in sedimentary basins. This model is applied to the neogene Vallès-Penedès half-graben (western Mediterranean, NE Spain) to show the capacity of the program when applied to a realistic geologic situation involving interactions between terrigenous clastics and carbonate sediments.

Power variation of some integral fractional processes

We consider the asymptotic behaviour of the realized power variation of processes of the form ¿t0usdBHs, where BH is a fractional Brownian motion with Hurst parameter H E(0,1), and u is a process with finite q-variation, q<1/(1¿H). We establish the stable convergence of the corresponding fluctuations. These results provide new statistical tools to study and detect the long-memory effect and the Hurst parameter.

On Ito's formula for elliptic diffusion processes

Bardina and Jolis [Stochastic process. Appl. 69 (1997) 83-109] prove an extension of Ito's formula for F(Xt, t), where F(x, t) has a locally square-integrable derivative in x that satisfies a mild continuity condition in t and X is a one-dimensional diffusion process such that the law of Xt has a density satisfying certain properties. This formula was expressed using quadratic covariation. Following the ideas of Eisenbaum [Potential Anal. 13 (2000) 303-328] concerning Brownian motion, we show that one can re-express this formula using integration over space and time with respect to local times in place of quadratic covariation. We also show that when the function F has a locally integrable derivative in t, we can avoid the mild continuity condition in t for the derivative of F in x.

Minimization of the experimental workload for the prediction of pollutants propagation in rivers. Mathematical modelling and knowledge re-use

The present thesis in focused on the minimization of experimental efforts for the prediction of pollutant propagation in rivers by mathematical modelling and knowledge re-use. Mathematical modelling is based on the well known advection-dispersion equation, while the knowledge re-use approach employs the methods of case based reasoning, graphical analysis and text mining. The thesis contribution to the pollutant transport research field consists of: (1) analytical and numerical models for pollutant transport prediction; (2) two novel techniques which enable the use of variable parameters along rivers in analytical models; (3) models for the estimation of pollutant transport characteristic parameters (velocity, dispersion coefficient and nutrient transformation rates) as functions of water flow, channel characteristics and/or seasonality; (4) the graphical analysis method to be used for the identification of pollution sources along rivers; (5) a case based reasoning tool for the identification of crucial information related to the pollutant transport modelling; (6) and the application of a software tool for the reuse of information during pollutants transport modelling research. These support tools are applicable in the water quality research field and in practice as well, as they can be involved in multiple activities. The models are capable of predicting pollutant propagation along rivers in case of both ordinary pollution and accidents. They can also be applied for other similar rivers in modelling of pollutant transport in rivers with low availability of experimental data concerning concentration. This is because models for parameter estimation developed in the present thesis enable the calculation of transport characteristic parameters as functions of river hydraulic parameters and/or seasonality. The similarity between rivers is assessed using case based reasoning tools, and additional necessary information can be identified by using the software for the information reuse. Such systems represent support for users and open up possibilities for new modelling methods, monitoring facilities and for better river water quality management tools. They are useful also for the estimation of environmental impact of possible technological changes and can be applied in the pre-design stage or/and in the practical use of processes as well.

A temperature predicting model for manufacturing processes requiring coiling

A model for predicting temperature evolution for automatic controling systems in manufacturing processes requiring the coiling of bars in the transfer table is presented. Although the method is of a general nature, the presentation in this work refers to the manufacturing of steel plates in hot rolling mills. The predicting strategy is based on a mathematical model of the evolution of temperature in a coiling and uncoiling bar and is presented in the form of a parabolic partial differential equation for a shape changing domain. The mathematical model is solved numerically by a space discretization via geometrically adaptive finite elements which accomodate the change in shape of the domain, using a computationally novel treatment of the resulting thermal contact problem due to coiling. Time is discretized according to a Crank-Nicolson scheme. Since the actual physical process takes less time than the time required by the process controlling computer to solve the full mathematical model, a special predictive device was developed, in the form of a set of least squares polynomials, based on the off-line numerical solution of the mathematical model.

Socially shared metacognitive regulation during collaborative learning processes in student dyads and small groups

Traditionally metacognition has been theorised, methodologically studied and empirically tested from the standpoint mainly of individuals and their learning contexts. In this dissertation the emergence of metacognition is analysed more broadly. The aim of the dissertation was to explore socially shared metacognitive regulation (SSMR) as part of collaborative learning processes taking place in student dyads and small learning groups. The specific aims were to extend the concept of individual metacognition to SSMR, to develop methods to capture and analyse SSMR and to validate the usefulness of the concept of SSMR in two different learning contexts; in face-to-face student dyads solving mathematical word problems and also in small groups taking part in inquiry-based science learning in an asynchronous computer-supported collaborative learning (CSCL) environment. This dissertation is comprised of four studies. In Study I, the main aim was to explore if and how metacognition emerges during problem solving in student dyads and then to develop a method for analysing the social level of awareness, monitoring, and regulatory processes emerging during the problem solving. Two dyads comprised of 10-year-old students who were high-achieving especially in mathematical word problem solving and reading comprehension were involved in the study. An in-depth case analysis was conducted. Data consisted of over 16 (30–45 minutes) videotaped and transcribed face-to-face sessions. The dyads solved altogether 151 mathematical word problems of different difficulty levels in a game-format learning environment. The interaction flowchart was used in the analysis to uncover socially shared metacognition. Interviews (also stimulated recall interviews) were conducted in order to obtain further information about socially shared metacognition. The findings showed the emergence of metacognition in a collaborative learning context in a way that cannot solely be explained by individual conception. The concept of socially-shared metacognition (SSMR) was proposed. The results highlighted the emergence of socially shared metacognition specifically in problems where dyads encountered challenges. Small verbal and nonverbal signals between students also triggered the emergence of socially shared metacognition. Additionally, one dyad implemented a system whereby they shared metacognitive regulation based on their strengths in learning. Overall, the findings suggested that in order to discover patterns of socially shared metacognition, it is important to investigate metacognition over time. However, it was concluded that more research on socially shared metacognition, from larger data sets, is needed. These findings formed the basis of the second study. In Study II, the specific aim was to investigate whether socially shared metacognition can be reliably identified from a large dataset of collaborative face-to-face mathematical word problem solving sessions by student dyads. We specifically examined different difficulty levels of tasks as well as the function and focus of socially shared metacognition. Furthermore, the presence of observable metacognitive experiences at the beginning of socially shared metacognition was explored. Four dyads participated in the study. Each dyad was comprised of high-achieving 10-year-old students, ranked in the top 11% of their fourth grade peers (n=393). Dyads were from the same data set as in Study I. The dyads worked face-to-face in a computer-supported, game-format learning environment. Problem-solving processes for 251 tasks at three difficulty levels taking place during 56 (30–45 minutes) lessons were video-taped and analysed. Baseline data for this study were 14 675 turns of transcribed verbal and nonverbal behaviours observed in four study dyads. The micro-level analysis illustrated how participants moved between different channels of communication (individual and interpersonal). The unit of analysis was a set of turns, referred to as an ‘episode’. The results indicated that socially shared metacognition and its function and focus, as well as the appearance of metacognitive experiences can be defined in a reliable way from a larger data set by independent coders. A comparison of the different difficulty levels of the problems suggested that in order to trigger socially shared metacognition in small groups, the problems should be more difficult, as opposed to moderately difficult or easy. Although socially shared metacognition was found in collaborative face-to-face problem solving among high-achieving student dyads, more research is needed in different contexts. This consideration created the basis of the research on socially shared metacognition in Studies III and IV. In Study III, the aim was to expand the research on SSMR from face-to-face mathematical problem solving in student dyads to inquiry-based science learning among small groups in an asynchronous computer-supported collaborative learning (CSCL) environment. The specific aims were to investigate SSMR’s evolvement and functions in a CSCL environment and to explore how SSMR emerges at different phases of the inquiry process. Finally, individual student participation in SSMR during the process was studied. An in-depth explanatory case study of one small group of four girls aged 12 years was carried out. The girls attended a class that has an entrance examination and conducts a language-enriched curriculum. The small group solved complex science problems in an asynchronous CSCL environment, participating in research-like processes of inquiry during 22 lessons (á 45–minute). Students’ network discussion were recorded in written notes (N=640) which were used as study data. A set of notes, referred to here as a ‘thread’, was used as the unit of analysis. The inter-coder agreement was regarded as substantial. The results indicated that SSMR emerges in a small group’s asynchronous CSCL inquiry process in the science domain. Hence, the results of Study III were in line with the previous Study I and Study II and revealed that metacognition cannot be reduced to the individual level alone. The findings also confirm that SSMR should be examined as a process, since SSMR can evolve during different phases and that different SSMR threads overlapped and intertwined. Although the classification of SSMR’s functions was applicable in the context of CSCL in a small group, the dominant function was different in the asynchronous CSCL inquiry in the small group in a science activity than in mathematical word problem solving among student dyads (Study II). Further, the use of different analytical methods provided complementary findings about students’ participation in SSMR. The findings suggest that it is not enough to code just a single written note or simply to examine who has the largest number of notes in the SSMR thread but also to examine the connections between the notes. As the findings of the present study are based on an in-depth analysis of a single small group, further cases were examined in Study IV, as well as looking at the SSMR’s focus, which was also studied in a face-to-face context. In Study IV, the general aim was to investigate the emergence of SSMR with a larger data set from an asynchronous CSCL inquiry process in small student groups carrying out science activities. The specific aims were to study the emergence of SSMR in the different phases of the process, students’ participation in SSMR, and the relation of SSMR’s focus to the quality of outcomes, which was not explored in previous studies. The participants were 12-year-old students from the same class as in Study III. Five small groups consisting of four students and one of five students (N=25) were involved in the study. The small groups solved ill-defined science problems in an asynchronous CSCL environment, participating in research-like processes of inquiry over a total period of 22 hours. Written notes (N=4088) detailed the network discussions of the small groups and these constituted the study data. With these notes, SSMR threads were explored. As in Study III, the thread was used as the unit of analysis. In total, 332 notes were classified as forming 41 SSMR threads. Inter-coder agreement was assessed by three coders in the different phases of the analysis and found to be reliable. Multiple methods of analysis were used. Results showed that SSMR emerged in all the asynchronous CSCL inquiry processes in the small groups. However, the findings did not reveal any significantly changing trend in the emergence of SSMR during the process. As a main trend, the number of notes included in SSMR threads differed significantly in different phases of the process and small groups differed from each other. Although student participation was seen as highly dispersed between the students, there were differences between students and small groups. Furthermore, the findings indicated that the amount of SSMR during the process or participation structure did not explain the differences in the quality of outcomes for the groups. Rather, when SSMRs were focused on understanding and procedural matters, it was associated with achieving high quality learning outcomes. In turn, when SSMRs were focused on incidental and procedural matters, it was associated with low level learning outcomes. Hence, the findings imply that the focus of any emerging SSMR is crucial to the quality of the learning outcomes. Moreover, the findings encourage the use of multiple research methods for studying SSMR. In total, the four studies convincingly indicate that a phenomenon of socially shared metacognitive regulation also exists. This means that it was possible to define the concept of SSMR theoretically, to investigate it methodologically and to validate it empirically in two different learning contexts across dyads and small groups. In-depth micro-level case analysis in Studies I and III showed the possibility to capture and analyse in detail SSMR during the collaborative process, while in Studies II and IV, the analysis validated the emergence of SSMR in larger data sets. Hence, validation was tested both between two environments and within the same environments with further cases. As a part of this dissertation, SSMR’s detailed functions and foci were revealed. Moreover, the findings showed the important role of observable metacognitive experiences as the starting point of SSMRs. It was apparent that problems dealt with by the groups should be rather difficult if SSMR is to be made clearly visible. Further, individual students’ participation was found to differ between students and groups. The multiple research methods employed revealed supplementary findings regarding SSMR. Finally, when SSMR was focused on understanding and procedural matters, this was seen to lead to higher quality learning outcomes. Socially shared metacognition regulation should therefore be taken into consideration in students’ collaborative learning at school similarly to how an individual’s metacognition is taken into account in individual learning.

Mathematical modeling of microwave dried celery leaves and determination of the effective moisture diffusivities and activation energy

Celery (Apium graveolens L. var. secalinum Alef) leaves with 50±0.07 g weight and 91.75±0.15% humidity (~11.21 db) were dried using 8 different microwave power densities ranging between 1.8-20 W g-1, until the humidity fell down to 8.95±0.23% (~0.1 db). Microwave drying processes were completed between 5.5 and 77 min depending on the microwave power densities. In this study, measured values were compared with predicted values obtained from twenty thin layer drying theoretical, semi-empirical and empirical equations with a new thin layer drying equation. Within applied microwave power density; models whose coefficient and correlation (R²) values are highest were chosen as the best models. Weibull distribution model gave the most suitable predictions at all power density. At increasing microwave power densities, the effective moisture diffusivity values ranged from 1.595 10-10 to 6.377 10-12 m2 s-1. The activation energy was calculated using an exponential expression based on Arrhenius equation. The linear relationship between the drying rate constant and effective moisture diffusivity gave the best fit.

Markovian Processes, Two-Sided Autoregressions and Finite-Sample Inference for Stationary and Nonstationary Autoregressive Processes.

In this paper, we develop finite-sample inference procedures for stationary and nonstationary autoregressive (AR) models. The method is based on special properties of Markov processes and a split-sample technique. The results on Markovian processes (intercalary independence and truncation) only require the existence of conditional densities. They are proved for possibly nonstationary and/or non-Gaussian multivariate Markov processes. In the context of a linear regression model with AR(1) errors, we show how these results can be used to simplify the distributional properties of the model by conditioning a subset of the data on the remaining observations. This transformation leads to a new model which has the form of a two-sided autoregression to which standard classical linear regression inference techniques can be applied. We show how to derive tests and confidence sets for the mean and/or autoregressive parameters of the model. We also develop a test on the order of an autoregression. We show that a combination of subsample-based inferences can improve the performance of the procedure. An application to U.S. domestic investment data illustrates the method.

Quadratic M-Estimators for ARCH-Type Processes

This paper addresses the issue of estimating semiparametric time series models specified by their conditional mean and conditional variance. We stress the importance of using joint restrictions on the mean and variance. This leads us to take into account the covariance between the mean and the variance and the variance of the variance, that is, the skewness and kurtosis. We establish the direct links between the usual parametric estimation methods, namely, the QMLE, the GMM and the M-estimation. The ususal univariate QMLE is, under non-normality, less efficient than the optimal GMM estimator. However, the bivariate QMLE based on the dependent variable and its square is as efficient as the optimal GMM one. A Monte Carlo analysis confirms the relevance of our approach, in particular, the importance of skewness.

Some Applications of Markov Additive Processes as Models in Insurance and Financial Mathematics

Cette thèse est principalement constituée de trois articles traitant des processus markoviens additifs, des processus de Lévy et d'applications en finance et en assurance. Le premier chapitre est une introduction aux processus markoviens additifs (PMA), et une présentation du problème de ruine et de notions fondamentales des mathématiques financières. Le deuxième chapitre est essentiellement l'article "Lévy Systems and the Time Value of Ruin for Markov Additive Processes" écrit en collaboration avec Manuel Morales et publié dans la revue European Actuarial Journal. Cet article étudie le problème de ruine pour un processus de risque markovien additif. Une identification de systèmes de Lévy est obtenue et utilisée pour donner une expression de l'espérance de la fonction de pénalité actualisée lorsque le PMA est un processus de Lévy avec changement de régimes. Celle-ci est une généralisation des résultats existant dans la littérature pour les processus de risque de Lévy et les processus de risque markoviens additifs avec sauts "phase-type". Le troisième chapitre contient l'article "On a Generalization of the Expected Discounted Penalty Function to Include Deficits at and Beyond Ruin" qui est soumis pour publication. Cet article présente une extension de l'espérance de la fonction de pénalité actualisée pour un processus subordinateur de risque perturbé par un mouvement brownien. Cette extension contient une série de fonctions escomptée éspérée des minima successives dus aux sauts du processus de risque après la ruine. Celle-ci a des applications importantes en gestion de risque et est utilisée pour déterminer la valeur espérée du capital d'injection actualisé. Finallement, le quatrième chapitre contient l'article "The Minimal entropy martingale measure (MEMM) for a Markov-modulated exponential Lévy model" écrit en collaboration avec Romuald Hervé Momeya et publié dans la revue Asia-Pacific Financial Market. Cet article présente de nouveaux résultats en lien avec le problème de l'incomplétude dans un marché financier où le processus de prix de l'actif risqué est décrit par un modèle exponentiel markovien additif. Ces résultats consistent à charactériser la mesure martingale satisfaisant le critère de l'entropie. Cette mesure est utilisée pour calculer le prix d'une option, ainsi que des portefeuilles de couverture dans un modèle exponentiel de Lévy avec changement de régimes.