Дисертації з теми "Differential probability"

Dans ce travail, nous présentons quelques exemples des effets du bruit sur la solution d'une équation aux dérivées partielles (EDP) dans trois contextes différents. Nous exam- inons d'abord deux équations aux dérivées partielles non linéaires dispersives, l'équation de Schrödinger non linéaire et l'équation de Korteweg - de Vries. Nous allons analyser les effets d'une condition initiale aléatoire sur certaines solutions spéciales, les solitons. Le deuxième cas considéré est une EDP linéaire, l'équation d'onde, avec conditions initiales aléatoires. Nous allons montrer qu'avec des conditions initiales aléatoires particulières c'est possible de réduire considérablement les coûts de stockage des données et de calcul d'un algorithme pour résoudre un problème inverse basé sur les mesures de la solution de cette équation au bord du domaine. Enfin, le troisième exemple considéré est celui de l'équation de transport linéaire avec un terme de dérive singulière. Nous allons montrer que l'ajout d'un terme de bruit multiplicatif interdit l'explosion des solutions, et cela sous des hypothèses très faibles pour lesquelles dans le cas déterministe on peut avoir l'explosion de la solution à temps fini.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 79-80).
This thesis consists of two parts. The first part applies a probabilistic approach to the study of the Wright-Fisher equation, an equation which is used to model demographic evolution in the presence of diffusion. The fundamental solution to the Wright-Fisher equation is carefully analyzed by relating it to the fundamental solution to a model equation which has the same degeneracy at one boundary. Estimates are given for short time behavior of the fundamental solution as well as its derivatives near the boundary. The second part studies the probabilistic extensions of the classical Cauchy functional equation for additive functions both in finite and infinite dimensions. The connection between additivity and linearity is explored under different circumstances, and the techniques developed in the process lead to results about the structure of abstract Wiener spaces. Both parts are joint work with Daniel W. Stroock.
by Linan Chen.
Ph.D.

[EN] Mathematical models based on deterministic differential equations do not take into account the inherent uncertainty of the physical phenomenon (in a wide sense) under study. In addition, inaccuracies in the collected data often arise due to errors in the measurements. It thus becomes necessary to treat the input parameters of the model as random quantities, in the form of random variables or stochastic processes. This gives rise to the study of random ordinary and partial differential equations. The computation of the probability density function of the stochastic solution is important for uncertainty quantification of the model output. Although such computation is a difficult objective in general, certain stochastic expansions for the model coefficients allow faithful representations for the stochastic solution, which permits approximating its density function. In this regard, Karhunen-Loève and generalized polynomial chaos expansions become powerful tools for the density approximation. Also, methods based on discretizations from finite difference numerical schemes permit approximating the stochastic solution, therefore its probability density function. The main part of this dissertation aims at approximating the probability density function of important mathematical models with uncertainties in their formulation. Specifically, in this thesis we study, in the stochastic sense, the following models that arise in different scientific areas: in Physics, the model for the damped pendulum; in Biology and Epidemiology, the models for logistic growth and Bertalanffy, as well as epidemiological models; and in Thermodynamics, the heat partial differential equation. We rely on Karhunen-Loève and generalized polynomial chaos expansions and on finite difference schemes for the density approximation of the solution. These techniques are only applicable when we have a forward model in which the input parameters have certain probability distributions already set. When the model coefficients are estimated from collected data, we have an inverse problem. The Bayesian inference approach allows estimating the probability distribution of the model parameters from their prior probability distribution and the likelihood of the data. Uncertainty quantification for the model output is then carried out using the posterior predictive distribution. In this regard, the last part of the thesis shows the estimation of the distributions of the model parameters from experimental data on bacteria growth. To do so, a hybrid method that combines Bayesian parameter estimation and generalized polynomial chaos expansions is used.
[ES] Los modelos matemáticos basados en ecuaciones diferenciales deterministas no tienen en cuenta la incertidumbre inherente del fenómeno físico (en un sentido amplio) bajo estudio. Además, a menudo se producen inexactitudes en los datos recopilados debido a errores en las mediciones. Por lo tanto, es necesario tratar los parámetros de entrada del modelo como cantidades aleatorias, en forma de variables aleatorias o procesos estocásticos. Esto da lugar al estudio de las ecuaciones diferenciales aleatorias. El cálculo de la función de densidad de probabilidad de la solución estocástica es importante en la cuantificación de la incertidumbre de la respuesta del modelo. Aunque dicho cálculo es un objetivo difícil en general, ciertas expansiones estocásticas para los coeficientes del modelo dan lugar a representaciones fieles de la solución estocástica, lo que permite aproximar su función de densidad. En este sentido, las expansiones de Karhunen-Loève y de caos polinomial generalizado constituyen herramientas para dicha aproximación de la densidad. Además, los métodos basados en discretizaciones de esquemas numéricos de diferencias finitas permiten aproximar la solución estocástica, por lo tanto, su función de densidad de probabilidad. La parte principal de esta disertación tiene como objetivo aproximar la función de densidad de probabilidad de modelos matemáticos importantes con incertidumbre en su formulación. Concretamente, en esta memoria se estudian, en un sentido estocástico, los siguientes modelos que aparecen en diferentes áreas científicas: en Física, el modelo del péndulo amortiguado; en Biología y Epidemiología, los modelos de crecimiento logístico y de Bertalanffy, así como modelos de tipo epidemiológico; y en Termodinámica, la ecuación en derivadas parciales del calor. Utilizamos expansiones de Karhunen-Loève y de caos polinomial generalizado y esquemas de diferencias finitas para la aproximación de la densidad de la solución. Estas técnicas solo son aplicables cuando tenemos un modelo directo en el que los parámetros de entrada ya tienen determinadas distribuciones de probabilidad establecidas. Cuando los coeficientes del modelo se estiman a partir de los datos recopilados, tenemos un problema inverso. El enfoque de inferencia Bayesiana permite estimar la distribución de probabilidad de los parámetros del modelo a partir de su distribución de probabilidad previa y la verosimilitud de los datos. La cuantificación de la incertidumbre para la respuesta del modelo se lleva a cabo utilizando la distribución predictiva a posteriori. En este sentido, la última parte de la tesis muestra la estimación de las distribuciones de los parámetros del modelo a partir de datos experimentales sobre el crecimiento de bacterias. Para hacerlo, se utiliza un método híbrido que combina la estimación de parámetros Bayesianos y los desarrollos de caos polinomial generalizado.
[CAT] Els models matemàtics basats en equacions diferencials deterministes no tenen en compte la incertesa inherent al fenomen físic (en un sentit ampli) sota estudi. A més a més, sovint es produeixen inexactituds en les dades recollides a causa d'errors de mesurament. Es fa així necessari tractar els paràmetres d'entrada del model com a quantitats aleatòries, en forma de variables aleatòries o processos estocàstics. Açò dóna lloc a l'estudi de les equacions diferencials aleatòries. El càlcul de la funció de densitat de probabilitat de la solució estocàstica és important per a quantificar la incertesa de la sortida del model. Tot i que, en general, aquest càlcul és un objectiu difícil d'assolir, certes expansions estocàstiques dels coeficients del model donen lloc a representacions fidels de la solució estocàstica, el que permet aproximar la seua funció de densitat. En aquest sentit, les expansions de Karhunen-Loève i de caos polinomial generalitzat esdevenen eines per a l'esmentada aproximació de la densitat. A més a més, els mètodes basats en discretitzacions mitjançant esquemes numèrics de diferències finites permeten aproximar la solució estocàstica, per tant la seua funció de densitat de probabilitat. La part principal d'aquesta dissertació té com a objectiu aproximar la funció de densitat de probabilitat d'importants models matemàtics amb incerteses en la seua formulació. Concretament, en aquesta memòria s'estudien, en un sentit estocàstic, els següents models que apareixen en diferents àrees científiques: en Física, el model del pèndol amortit; en Biologia i Epidemiologia, els models de creixement logístic i de Bertalanffy, així com models de tipus epidemiològic; i en Termodinàmica, l'equació en derivades parcials de la calor. Per a l'aproximació de la densitat de la solució, ens basem en expansions de Karhunen-Loève i de caos polinomial generalitzat i en esquemes de diferències finites. Aquestes tècniques només són aplicables quan tenim un model cap avant en què els paràmetres d'entrada tenen ja determinades distribucions de probabilitat. Quan els coeficients del model s'estimen a partir de les dades recollides, tenim un problema invers. L'enfocament de la inferència Bayesiana permet estimar la distribució de probabilitat dels paràmetres del model a partir de la seua distribució de probabilitat prèvia i la versemblança de les dades. La quantificació de la incertesa per a la resposta del model es fa mitjançant la distribució predictiva a posteriori. En aquest sentit, l'última part de la tesi mostra l'estimació de les distribucions dels paràmetres del model a partir de dades experimentals sobre el creixement de bacteris. Per a fer-ho, s'utilitza un mètode híbrid que combina l'estimació de paràmetres Bayesiana i els desenvolupaments de caos polinomial generalitzat.
This work has been supported by the Spanish Ministerio de Econom´ıa y Competitividad grant MTM2017–89664–P.
Calatayud Gregori, J. (2020). Computational methods for random differential equations: probability density function and estimation of the parameters [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/138396
TESIS
Premiado

In this study, we aim to develop new likelihood based method for estimating parameters of ordinary differential equations (ODEs) / delay differential equations (DDEs) models. Those models are important for modeling dynamical processes that are described in terms of their derivatives and are widely used in many fields of modern science, such as physics, chemistry, biology and social sciences. We use our new approach to study a distributed delay differential equation model, the statistical inference of which has been unexplored, to our knowledge. Estimating a distributed DDE model or ODE model with time varying coefficients results in a large number of parameters. We also apply regularization for efficient estimation of such models. We assess the performance of our new approaches using simulation and applied them to analyzing data from epidemiology and ecology.

This study was designed to test the effectiveness of 2 different behavioral interventions: a high-probability request sequence and a differential reinforcement of alternative behaviors (DRA) procedure in a classroom setting. The aim of the interventions was to reduce the frequency of task refusal as well as increase the frequency of task compliance in adolescents in a general education setting. The study included 4 adolescents with the same teacher who were reported by their teacher as completing 50% or less of their course work since the beginning of the school year. The teacher implemented the interventions with the participants to test their potential effectiveness. Each student responded differently to the interventions. This was demonstrated using visual analysis of graphs as well as a comparison of descriptive statistics. Some were more compliant when the teacher implemented the high-probability request sequence; others demonstrated greater compliance with the DRA in place. Two participants also demonstrated higher levels of compliance beginning with placement of a camera (and operator) prior to the high-probability request sequence or the DRA implementation. These results indicate that each of these interventions may have the potential to increase compliance with classroom tasks for typically functioning adolescents through the mechanism of increased attention.

[EN] This thesis concerns the analysis of differential equations with uncertain input parameters, in the form of random variables or stochastic processes with any type of probability distributions. In modeling, the input coefficients are set from experimental data, which often involve uncertainties from measurement errors. Moreover, the behavior of the physical phenomenon under study does not follow strict deterministic laws. It is thus more realistic to consider mathematical models with randomness in their formulation. The solution, considered in the sample-path or the mean square sense, is a smooth stochastic process, whose uncertainty has to be quantified. Uncertainty quantification is usually performed by computing the main statistics (expectation and variance) and, if possible, the probability density function. In this dissertation, we study random linear models, based on ordinary differential equations with and without delay and on partial differential equations. The linear structure of the models makes it possible to seek for certain probabilistic solutions and even approximate their probability density functions, which is a difficult goal in general. A very important part of the dissertation is devoted to random second-order linear differential equations, where the coefficients of the equation are stochastic processes and the initial conditions are random variables. The study of this class of differential equations in the random setting is mainly motivated because of their important role in Mathematical Physics. We start by solving the randomized Legendre differential equation in the mean square sense, which allows the approximation of the expectation and the variance of the stochastic solution. The methodology is extended to general random second-order linear differential equations with analytic (expressible as random power series) coefficients, by means of the so-called Fröbenius method. A comparative case study is performed with spectral methods based on polynomial chaos expansions. On the other hand, the Fröbenius method together with Monte Carlo simulation are used to approximate the probability density function of the solution. Several variance reduction methods based on quadrature rules and multilevel strategies are proposed to speed up the Monte Carlo procedure. The last part on random second-order linear differential equations is devoted to a random diffusion-reaction Poisson-type problem, where the probability density function is approximated using a finite difference numerical scheme. The thesis also studies random ordinary differential equations with discrete constant delay. We study the linear autonomous case, when the coefficient of the non-delay component and the parameter of the delay term are both random variables while the initial condition is a stochastic process. It is proved that the deterministic solution constructed with the method of steps that involves the delayed exponential function is a probabilistic solution in the Lebesgue sense. Finally, the last chapter is devoted to the linear advection partial differential equation, subject to stochastic velocity field and initial condition. We solve the equation in the mean square sense and provide new expressions for the probability density function of the solution, even in the non-Gaussian velocity case.
[ES] Esta tesis trata el análisis de ecuaciones diferenciales con parámetros de entrada aleatorios, en la forma de variables aleatorias o procesos estocásticos con cualquier tipo de distribución de probabilidad. En modelización, los coeficientes de entrada se fijan a partir de datos experimentales, los cuales suelen acarrear incertidumbre por los errores de medición. Además, el comportamiento del fenómeno físico bajo estudio no sigue patrones estrictamente deterministas. Es por tanto más realista trabajar con modelos matemáticos con aleatoriedad en su formulación. La solución, considerada en el sentido de caminos aleatorios o en el sentido de media cuadrática, es un proceso estocástico suave, cuya incertidumbre se tiene que cuantificar. La cuantificación de la incertidumbre es a menudo llevada a cabo calculando los principales estadísticos (esperanza y varianza) y, si es posible, la función de densidad de probabilidad. En este trabajo, estudiamos modelos aleatorios lineales, basados en ecuaciones diferenciales ordinarias con y sin retardo, y en ecuaciones en derivadas parciales. La estructura lineal de los modelos nos permite buscar ciertas soluciones probabilísticas e incluso aproximar su función de densidad de probabilidad, lo cual es un objetivo complicado en general. Una parte muy importante de la disertación se dedica a las ecuaciones diferenciales lineales de segundo orden aleatorias, donde los coeficientes de la ecuación son procesos estocásticos y las condiciones iniciales son variables aleatorias. El estudio de esta clase de ecuaciones diferenciales en el contexto aleatorio está motivado principalmente por su importante papel en la Física Matemática. Empezamos resolviendo la ecuación diferencial de Legendre aleatorizada en el sentido de media cuadrática, lo que permite la aproximación de la esperanza y la varianza de la solución estocástica. La metodología se extiende al caso general de ecuaciones diferenciales lineales de segundo orden aleatorias con coeficientes analíticos (expresables como series de potencias), mediante el conocido método de Fröbenius. Se lleva a cabo un estudio comparativo con métodos espectrales basados en expansiones de caos polinomial. Por otro lado, el método de Fröbenius junto con la simulación de Monte Carlo se utilizan para aproximar la función de densidad de probabilidad de la solución. Para acelerar el procedimiento de Monte Carlo, se proponen varios métodos de reducción de la varianza basados en reglas de cuadratura y estrategias multinivel. La última parte sobre ecuaciones diferenciales lineales de segundo orden aleatorias estudia un problema aleatorio de tipo Poisson de difusión-reacción, en el que la función de densidad de probabilidad es aproximada mediante un esquema numérico de diferencias finitas. En la tesis también se tratan ecuaciones diferenciales ordinarias aleatorias con retardo discreto y constante. Estudiamos el caso lineal y autónomo, cuando el coeficiente de la componente no retardada i el parámetro del término retardado son ambos variables aleatorias mientras que la condición inicial es un proceso estocástico. Se demuestra que la solución determinista construida con el método de los pasos y que involucra la función exponencial retardada es una solución probabilística en el sentido de Lebesgue. Finalmente, el último capítulo lo dedicamos a la ecuación en derivadas parciales lineal de advección, sujeta a velocidad y condición inicial estocásticas. Resolvemos la ecuación en el sentido de media cuadrática y damos nuevas expresiones para la función de densidad de probabilidad de la solución, incluso en el caso de velocidad no Gaussiana.
[CAT] Aquesta tesi tracta l'anàlisi d'equacions diferencials amb paràmetres d'entrada aleatoris, en la forma de variables aleatòries o processos estocàstics amb qualsevol mena de distribució de probabilitat. En modelització, els coeficients d'entrada són fixats a partir de dades experimentals, les quals solen comportar incertesa pels errors de mesurament. A més a més, el comportament del fenomen físic sota estudi no segueix patrons estrictament deterministes. És per tant més realista treballar amb models matemàtics amb aleatorietat en la seua formulació. La solució, considerada en el sentit de camins aleatoris o en el sentit de mitjana quadràtica, és un procés estocàstic suau, la incertesa del qual s'ha de quantificar. La quantificació de la incertesa és sovint duta a terme calculant els principals estadístics (esperança i variància) i, si es pot, la funció de densitat de probabilitat. En aquest treball, estudiem models aleatoris lineals, basats en equacions diferencials ordinàries amb retard i sense, i en equacions en derivades parcials. L'estructura lineal dels models ens fa possible cercar certes solucions probabilístiques i inclús aproximar la seua funció de densitat de probabilitat, el qual és un objectiu complicat en general. Una part molt important de la dissertació es dedica a les equacions diferencials lineals de segon ordre aleatòries, on els coeficients de l'equació són processos estocàstics i les condicions inicials són variables aleatòries. L'estudi d'aquesta classe d'equacions diferencials en el context aleatori està motivat principalment pel seu important paper en Física Matemàtica. Comencem resolent l'equació diferencial de Legendre aleatoritzada en el sentit de mitjana quadràtica, el que permet l'aproximació de l'esperança i la variància de la solució estocàstica. La metodologia s'estén al cas general d'equacions diferencials lineals de segon ordre aleatòries amb coeficients analítics (expressables com a sèries de potències), per mitjà del conegut mètode de Fröbenius. Es duu a terme un estudi comparatiu amb mètodes espectrals basats en expansions de caos polinomial. Per altra banda, el mètode de Fröbenius juntament amb la simulació de Monte Carlo són emprats per a aproximar la funció de densitat de probabilitat de la solució. Per a accelerar el procediment de Monte Carlo, es proposen diversos mètodes de reducció de la variància basats en regles de quadratura i estratègies multinivell. L'última part sobre equacions diferencials lineals de segon ordre aleatòries estudia un problema aleatori de tipus Poisson de difusió-reacció, en què la funció de densitat de probabilitat és aproximada mitjançant un esquema numèric de diferències finites. En la tesi també es tracten equacions diferencials ordinàries aleatòries amb retard discret i constant. Estudiem el cas lineal i autònom, quan el coeficient del component no retardat i el paràmetre del terme retardat són ambdós variables aleatòries mentre que la condició inicial és un procés estocàstic. Es prova que la solució determinista construïda amb el mètode dels passos i que involucra la funció exponencial retardada és una solució probabilística en el sentit de Lebesgue. Finalment, el darrer capítol el dediquem a l'equació en derivades parcials lineal d'advecció, subjecta a velocitat i condició inicial estocàstiques. Resolem l'equació en el sentit de mitjana quadràtica i donem noves expressions per a la funció de densitat de probabilitat de la solució, inclús en el cas de velocitat no Gaussiana.
This work has been supported by the Spanish Ministerio de Economía y Competitividad grant MTM2017–89664–P. I acknowledge the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València.
Jornet Sanz, M. (2020). Mean square solutions of random linear models and computation of their probability density function [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/138394
TESIS

Cette thèse porte sur l'étude des dynamiques spatiales et évolutives d'une population à l'aide d'outils probabilistes et déterministes. Dans la première partie, nous cherchons à comprendre l'effet de l'hétérogénéité de l'environnement sur l'évolution des espèces. La population considérée est modélisée par un processus individu-centré avec interactions qui décrit les événements de naissances, morts, mutations et diffusions spatiales de chaque individu. Les taux des événements dépendent des caractéristiques des individus : traits phénotypes et positions spatiales. Dans un premier temps, nous étudions le système d'équations aux dérivées partielles qui décrit la dynamique spatiale et démographique d'une population composée de deux traits dans une limite grande population. Nous caractérisons précisément les conditions d'extinction et de survie en temps long de cette population. Dans un deuxième temps, nous étudions le modèle individuel initial sous deux asymptotiques : grande population et mutations rares de telle sorte que les échelles de temps démographiques et mutationnelles sont séparées. Ainsi, lorsqu'un mutant apparaît, la population résidente est à l'équilibre démographique. Nous cherchons alors à caractériser la probabilité de survie de la population issue de ce mutant. Puis, en étudiantle processus dans l'échelle des mutations, nous prouvons que le processus individu-centré converge vers un processus de sauts qui décrit les fixations successives des traits les plus avantagés ainsi que la répartition spatiale des populations portant ces traits. Nous généralisons ensuite le modèle pour introduire des interactions de type mutualiste entre deux espèces. Nous étudions ce modèle dans une limite de grande population. Nous donnons par ailleurs des résultats numériques et une analyse biologique détaillée des comportements obtenus autour de deux problématiques : la coévolution de niches spatiales et phénotypiques d'espèces en interaction mutualiste et les dynamiques d'invasions d'un espace homogène par des espèces mutualistes. Dans la deuxième partie de cette thèse, nous développons un modèle probabiliste pour étudier finement l'effet d'une préférence sexuelle sur la spéciation. La population est ici structurée sur deux patchs et les individus, caractérisés par un trait, sont écologiquement et démographiquement équivalents et se distinguent uniquement par leur préférence sexuelle: deux individus de même trait ont plus de chance de se reproduire que deux individus de traits distincts. Nous montrons qu'en l'absence de toute autre différence écologique, la préférence sexuelle mène à un isolement reproductif entre les deux patchs
We study the spatial and evolutionary dynamics of a population by using probabilistic and deterministic tools. In the first part of this thesis, we are concerned with the influence of a heterogeneous environment on the evolution of species. The population is modeled by an individual-based process with some interactions and which describes the birth, the death, the mutation and the spatial diffusion of each individual. The rates of those events depend on the characteristics of the individuals : their phenotypic trait and their spatial location. First, we study the system of partial differential equations that describes the spatial and demographic dynamics of a population composed of two traits in a large population limit. We characterize precisely the conditions of extinction and long time survival for this population. Secondly, we study the initial individual-based model under two asymptotic: large population and rare mutations such as demographic and mutational timescales are separated. Thus, when a mutant appears, the resident population has reached its demographic balance. We characterize the survival probability of the population descended from this mutant. Then, by studyingthe process in the mutational scale, we prove that the microscopic process converges to a jump process which describes the successive fixations of the most advantaged traits and the spatial distribution of populations carrying these traits. We then extend the model to introduce mutualistic interactions between two species. We study this model in a limit of large population. We also give numerical results and a detailed biological behavior analysis around two issues: the co-evolution of phenotypic and spatial niches of mutualistic species and the invasion dynamics of a homogeneous space by these species. In the second part of this thesis, we develop a probabilistic model to study the effect of the sexual preference on the speciation. Here, the population is structured on two patches and the individuals, characterized by a trait, are ecologically and demographically similar and differ only in their sexual preferences: two individuals of the same trait are more likely to reproduce than two individuals of distinct traits. We show that in the absence of any other ecological differences, the sexual preferences lead to reproductive isolation between the two patches

Desde las contribuciones de Isaac Newton, Gottfried Wilhelm Leibniz, Jacob y Johann Bernoulli en el siglo XVII hasta ahora, las ecuaciones en diferencias y las diferenciales han demostrado su capacidad para modelar satisfactoriamente problemas complejos de gran interés en Ingeniería, Física, Epidemiología, etc. Pero, desde un punto de vista práctico, los parámetros o inputs (condiciones iniciales/frontera, término fuente y/o coeficientes), que aparecen en dichos problemas, son fijados a partir de ciertos datos, los cuales pueden contener un error de medida. Además, pueden existir factores externos que afecten al sistema objeto de estudio, de modo que su complejidad haga que no se conozcan de forma cierta los parámetros de la ecuación que modeliza el problema. Todo ello justifica considerar los parámetros de la ecuación en diferencias o de la ecuación diferencial como variables aleatorias o procesos estocásticos, y no como constantes o funciones deterministas, respectivamente. Bajo esta consideración aparecen las ecuaciones en diferencias y las ecuaciones diferenciales aleatorias. Esta tesis hace un recorrido resolviendo, desde un punto de vista probabilístico, distintos tipos de ecuaciones en diferencias y diferenciales aleatorias, aplicando fundamentalmente el método de Transformación de Variables Aleatorias. Esta técnica es una herramienta útil para la obtención de la función de densidad de probabilidad de un vector aleatorio, que es una transformación de otro vector aleatorio cuya función de densidad de probabilidad es conocida. En definitiva, el objetivo de este trabajo es el cálculo de la primera función de densidad de probabilidad del proceso estocástico solución en diversos problemas basados en ecuaciones en diferencias y diferenciales aleatorias. El interés por determinar la primera función de densidad de probabilidad se justifica porque dicha función determinista caracteriza la información probabilística unidimensional, como media, varianza, asimetría, curtosis, etc., de la solución de la ecuación en diferencias o diferencial correspondiente. También permite determinar la probabilidad de que acontezca un determinado suceso de interés que involucre a la solución. Además, en algunos casos, el estudio teórico realizado se completa mostrando su aplicación a problemas de modelización con datos reales, donde se aborda el problema de la estimación de distribuciones estadísticas paramétricas de los inputs en el contexto de las ecuaciones en diferencias y diferenciales aleatorias.
Ever since the early contributions by Isaac Newton, Gottfried Wilhelm Leibniz, Jacob and Johann Bernoulli in the XVII century until now, difference and differential equations have uninterruptedly demonstrated their capability to model successfully interesting complex problems in Engineering, Physics, Chemistry, Epidemiology, Economics, etc. But, from a practical standpoint, the application of difference or differential equations requires setting their inputs (coefficients, source term, initial and boundary conditions) using sampled data, thus containing uncertainty stemming from measurement errors. In addition, there are some random external factors which can affect to the system under study. Then, it is more advisable to consider input data as random variables or stochastic processes rather than deterministic constants or functions, respectively. Under this consideration random difference and differential equations appear. This thesis makes a trail by solving, from a probabilistic point of view, different types of random difference and differential equations, applying fundamentally the Random Variable Transformation method. This technique is an useful tool to obtain the probability density function of a random vector that results from mapping another random vector whose probability density function is known. Definitely, the goal of this dissertation is the computation of the first probability density function of the solution stochastic process in different problems, which are based on random difference or differential equations. The interest in determining the first probability density function is justified because this deterministic function characterizes the one-dimensional probabilistic information, as mean, variance, asymmetry, kurtosis, etc. of corresponding solution of a random difference or differential equation. It also allows to determine the probability of a certain event of interest that involves the solution. In addition, in some cases, the theoretical study carried out is completed, showing its application to modelling problems with real data, where the problem of parametric statistics distribution estimation is addressed in the context of random difference and differential equations.
Des de les contribucions de Isaac Newton, Gottfried Wilhelm Leibniz, Jacob i Johann Bernoulli al segle XVII fins a l'actualitat, les equacions en diferències i les diferencials han demostrat la seua capacitat per a modelar satisfactòriament problemes complexos de gran interés en Enginyeria, Física, Epidemiologia, etc. Però, des d'un punt de vista pràctic, els paràmetres o inputs (condicions inicials/frontera, terme font i/o coeficients), que apareixen en aquests problemes, són fixats a partir de certes dades, les quals poden contenir errors de mesura. A més, poden existir factors externs que afecten el sistema objecte d'estudi, de manera que, la seua complexitat faça que no es conega de forma certa els inputs de l'equació que modelitza el problema. Tot aço justifica la necessitat de considerar els paràmetres de l'equació en diferències o de la equació diferencial com a variables aleatòries o processos estocàstics, i no com constants o funcions deterministes. Sota aquesta consideració apareixen les equacions en diferències i les equacions diferencials aleatòries. Aquesta tesi fa un recorregut resolent, des d'un punt de vista probabilístic, diferents tipus d'equacions en diferències i diferencials aleatòries, aplicant fonamentalment el mètode de Transformació de Variables Aleatòries. Aquesta tècnica és una eina útil per a l'obtenció de la funció de densitat de probabilitat d'un vector aleatori, que és una transformació d'un altre vector aleatori i la funció de densitat de probabilitat és del qual és coneguda. En definitiva, l'objectiu d'aquesta tesi és el càlcul de la primera funció de densitat de probabilitat del procés estocàstic solució en diversos problemes basats en equacions en diferències i diferencials. L'interés per determinar la primera funció de densitat es justifica perquè aquesta funció determinista caracteritza la informació probabilística unidimensional, com la mitjana, variància, asimetria, curtosis, etc., de la solució de l'equació en diferències o l'equació diferencial aleatòria corresponent. També permet determinar la probabilitat que esdevinga un determinat succés d'interés que involucre la solució. A més, en alguns casos, l'estudi teòric realitzat es completa mostrant la seua aplicació a problemes de modelització amb dades reals, on s'aborda el problema de l'estimació de distribucions estadístiques paramètriques dels inputs en el context de les equacions en diferències i diferencials aleatòries.
Navarro Quiles, A. (2018). COMPUTATIONAL METHODS FOR RANDOM DIFFERENTIAL EQUATIONS: THEORY AND APPLICATIONS [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/98703
TESIS

This doctoral thesis is concerned with some theoretical and practical questions related to backward stochastic differential equations (BSDEs) and more specifically their connection with some parabolic partial differential equations (PDEs). The thesis is made of three parts. In the first part, we study the probabilistic representation for a class of multidimensional PDEs with quadratic nonlinearities of a special form. We obtain a representation formula for the PDE solution in terms of the solutions to a Lipschitz BSDE. We then use this representation to obtain an estimate on the gradient of the PDE solutions by probabilistic means. In the course of our analysis, we are led to prove some results for the associated multidimensional quadratic BSDEs, namely an existence result and a partial uniqueness result. In the second part, we study the well-posedness of a very general quadratic reflected BSDE driven by a continuous martingale. We obtain the comparison theorem, the special comparison theorem for reflected BSDEs (which allows to compare the increasing processes of two solutions), the uniqueness and existence of solutions, as well as a stability result. The comparison theorem (from which uniqueness follows) and the special comparison theorem are obtained through natural techniques and minimal assumptions. The existence is based on a perturbative procedure, and holds for a driver whis is Lipschitz, or slightly-superlinear, or monotone with arbitrary growth in y. Finally, we obtain a stability result, which gives in particular a local Lipschitz estimate in BMO for the martingale part of the solution. In the third and last part, we study the time-discretization of BSDEs having nonlinearities that are monotone but with polynomial growth in the primary variable. We show that in that case, the explicit Euler scheme is likely to diverge, while the implicit scheme converges. In fact, by studying the family of θ-schemes, which are mixed explicit-implicit, θ characterizing the degree of implicitness, we find that the scheme converges when the implicit component is dominant (θ ≥ 1/2 ). We then propose a tamed explicit scheme, which converges. We show that the implicit-dominant schemes with θ > 1/2 and our tamed explicit scheme converge with order 1/2 , while the trapezoidal scheme (θ = 1/2) converges with order 7/4.

We construct a mathematical model of an order driven market where traders can submit limit orders and market orders to buy and sell securities. We adapt the notion of no free lunch of Harrison and Kreps and Jouini and Kallal to our setting and we prove a no-arbitrage theorem for the model of the order driven market. Furthermore, we compute signatures of order books of different financial markets. Signatures, i.e. the full sequence of definite iterated integrals of a path, are one of the fundamental elements of the theory of rough paths. The theory of rough paths provides a framework to describe the evolution of dynamical systems that are driven by rough signals, including rough paths based on Brownian motion and fractional Brownian motion (see the work of Lyons). We show how we can obtain the solution of a polynomial differential equation and its (truncated) signature from the signature of the driving signal and the initial value. We also present and analyse an ODE method for the numerical solution of rough differential equations. We derive error estimates and we prove that it achieves the same rate of convergence as the corresponding higher order Euler schemes studied by Davie and Friz and Victoir. At the same time, it enhances stability. The method has been implemented for the case of polynomial vector fields as part of the CoRoPa software package which is available at http://coropa.sourceforge.net. We describe both the algorithm and the implementation and we show by giving examples how it can be used to compute the pathwise solution of stochastic rough differential equations driven by Brownian rough paths and fractional Brownian rough paths.

Stochastic analysis on fractals is, as one might expect, a subfield of analysis on fractals. An intuitive starting point is to observe that on many fractals, one can define diffusion processes whose law is in some sense invariant with respect to the symmetries and self-similarities of the fractal. These can be interpreted as fractal-valued counterparts of standard Brownian motion on Rd. One can study these diffusions directly, for example by computing heat kernel and hitting time estimates. On the other hand, by associating the infinitesimal generator of the fractal-valued diffusion with the Laplacian on Rd, it is possible to pose stochastic partial differential equations on the fractal such as the stochastic heat equation and stochastic wave equation. In this thesis we investigate a variety of questions concerning the properties of diffusions on fractals and the parabolic and hyperbolic SPDEs associated with them. Key results include an extension of Kolmogorov's continuity theorem to stochastic processes indexed by fractals, and existence and uniqueness of solutions to parabolic SPDEs on fractals with Lipschitz data.

Some recent works in the field of mathematical finance have brought new light on the importance of studying the regularity and the tail asymptotics of distributions for certain classes of diffusions with non-globally smooth coefficients. In this Ph.D. dissertation we deal with some issues in this framework. In a first part, we study the existence, smoothness and space asymptotics of densities for the solutions of stochastic differential equations assuming only local conditions on the coefficients of the equation. Our analysis is based on Malliavin calculus tools and on " tube estimates " for Ito processes, namely estimates for the probability that the trajectory of an Ito process remains close to a deterministic curve. We obtain significant estimates of densities and distribution functions in general classes of option pricing models, including generalisations of CIR and CEV processes and Local-Stochastic Volatility models. In the latter case, the estimates we derive have an impact on the moment explosion of the underlying price and, consequently, on the large-strike behaviour of the implied volatility. Parametric implied volatility modeling, in its turn, makes the object of the second part. In particular, we focus on J. Gatheral's SVI model, first proposing an effective quasi-explicit calibration procedure and displaying its performances on market data. Then, we analyse the capability of SVI to generate efficient approximations of symmetric smiles, building an explicit time-dependent parameterization. We provide and test the numerical application to the Heston model (without and with displacement), for which we generate semi-closed expressions of the smile

In this thesis, we present a new existence result for Lévy-type processes. Lévy-type processes behave locally like a Lévy process, but the Lévy triplet may depend on the current position of the process. They can be characterized by their so-called symbol; this is the analogue of the characteristic exponent in the Lévy case. Using a parametrix construction, we prove the existence of Lévy-type processes with a given symbol under weak regularity assumptions on the regularity of the symbol. Applications range from existence results for stable-like processes and mixed processes to uniqueness results for Lévy-driven stochastic differential equations. Moreover, we discuss sufficient conditions for the existence of moments of Lévy-type processes and derive estimates for fractional moments.

Being in the era of information technology, importance and applicability of analytical statistical model an interdisciplinary setting in the modern statistics have increased significantly. Conceptually understanding the vulnerabilities in statistical perspective helps to develop the set of modern statistical models and bridges the gap between cybersecurity and abstract statistical /mathematical knowledge. In this dissertation, our primary goal is to develop series of the strong statistical model in software vulnerability in conjunction with Common Vulnerability Scoring System (CVSS) framework. In nutshell, the overall research lies at the intersection of statistical modeling, cybersecurity, and data mining. Furthermore, we generalize the model of software vulnerability to health science particularly in the stomach cancer data. In the context of cybersecurity, we have applied the well-known Markovian process in the combination of CVSS framework to determine the overall network security risk. The developed model can be used to identify critical nodes in the host access graph where attackers may be most likely to focus. Based on that information, a network administrator can make appropriate, prioritized decisions for system patching. Further, a flexible risk ranking technique is described, where the decisions made by an attacker can be adjusted using a bias factor. The model can be generalized for use with complicated network environments. We have further proposed a vulnerability analytic prediction model based on linear and non-linear approaches via time series analysis. Using currently available data from National Vulnerability Database (NVD) this study develops and present sets of predictive model by utilizing Auto Regressive Moving Average (ARIMA), Artificial Neural Network (ANN), and Support Vector Machine (SVM) settings. The best model which provides the minimum error rate is selected for prediction of future vulnerabilities. In addition, we purpose a new philosophy of software vulnerability life cycle. It says that vulnerability saturation is a local phenomenon, and it possesses an increasing cyclic behavior within the software vulnerability life cycle. Based on the new philosophy of software vulnerability life cycle, we purpose new effective differential equation model to predict future software vulnerabilities by utilizing the vulnerability dataset of three major OS: Windows 7, Linux Kernel, and Mac OS X. The proposed analytical model is compared with existing models in terms of fitting and prediction accuracy. Finally, the predictive model not only applicable to predict future vulnerability but it can be used in the various domain such as engineering, finance, business, health science, and among others. For instance, we extended the idea on health science; to predict the malignant tumor size of stomach cancer as a function of age based on the given historical data from Surveillance Epidemiology and End Results (SEER).

In this thesis, we will develop the fundamental properties of financial mathematics, with a focus on establishing meaningful connections between martingale theory, stochastic calculus, and measure-theoretic probability. We first consider a simple binomial model in discrete time, and assume the impossibility of earning a riskless profit, known as arbitrage. Under this no-arbitrage assumption alone, we stumble upon a strange new probability measure Q, according to which every risky asset is expected to grow as though it were a bond. As it turns out, this measure Q also gives the arbitrage-free pricing formula for every asset on our market. In considering a slightly more complicated model over a finite probability space, we see that Q once again makes its appearance. Finally, in the context of continuous time, we build a framework of stochastic calculus to model the trajectories of asset prices on a finite time interval. Under the absence of arbitrage once more, we see that Q makes its return as a Radon-Nikodym derivative of our initial probability measure. Finally, we use the properties of Q and a stochastic differential equation that models the dynamics of the assets of our market, known as the Ito formula, in order to derive the classic Black-Scholes Equation.

The main motivation behind writing this thesis was to construct numerical methods to approximate solutions to differential equations driven by rough paths, where the solution is considered in the rough path-sense. Rough paths of inhomogeneous degree of smoothness as driving noise are considered. We also aimed to find applications of these numerical methods to stochastic differential equations. After sketching the core ideas of the Rough Paths Theory in Chapter 1, the versions of the core theorems corresponding to the inhomogeneous degree of smoothness case are stated and proved in Chapter 2 along with some auxiliary claims on the continuity of the solution in a certain sense, including an RDE-version of Gronwall's lemma. In Chapter 3, numerical schemes for approximating solutions to differential equations driven by rough paths of inhomogeneous degree of smoothness are constructed. We start with setting up some principles of approximations. Then a general class of local approximations is introduced. This class is used to construct global approximations by pasting together the local ones. A general sufficient condition on the local approximations implying global convergence is given and proved. The next step is to construct particular local approximations in finite dimensions based on solutions to ordinary differential equations derived locally and satisfying the sufficient condition for global convergence. These local approximations require strong conditions on the one-form defining the rough differential equation. Finally, we show that when the local ODE-based schemes are applied in combination with rough polynomial approximations, the conditions on the one-form can be weakened. In Chapter 4, the results of Gyurko & Lyons (2010) on path-wise approximation of solutions to stochastic differential equations are recalled and extended to the truncated signature level of the solution. Furthermore, some practical considerations related to the implementation of high order schemes are described. The effectiveness of the derived schemes is demonstrated on numerical examples. In Chapter 5, the background theory of the Kusuoka-Lyons-Victoir (KLV) family of weak approximations is recalled and linked to the results of Chapter 4. We highlight how the different versions of the KLV family are related. Finally, a numerical evaluation of the autonomous ODE-based versions of the family is carried out, focusing on SDEs in dimensions up to 4, using cubature formulas of different degrees and several high order numerical ODE solvers. We demonstrate the effectiveness and the occasional non-effectiveness of the numerical approximations in cases when the KLV family is used in its original version and also when used in combination with partial sampling methods (Monte-Carlo, TBBA) and Romberg extrapolation.

This thesis is about double product integrals with pseudo rotational generator, and aims to exhibit them as unitary implementors of Bogolubov transformations. We further introduce these concepts in this abstract and describe their roles in the thesis's chapters. The notion of product integral, (simple product integral, not double) is not a new one, but is unfamiliar to many a mathematician. Product integrals were first investigated by Volterra in the nineteenth century. Though often regarded as merely a notation for solutions of differential equations, they provide a priori a multiplicative analogue of the additive integration theories of Riemann, Stieltjes and Lebesgue. See Slavik [2007] for a historical overview of the subject. Extensions of the theory of product integrals to multiplicative versions of Ito and especially quantum Ito calculus were first studied by Hudson, Ion and Parthasarathy in the 1980's, Hudson et al. [1982]. The first developments of double product integrals was a theory of an algebraic kind developed by Hudson and Pulmannova motivated by the study of the solution of the quantum Yang-Baxter equation by the construction of quantum groups, see Hudson and Pulmaanova [2005]. This was a purely algebraic theory based on formal power series in a formal parameter. However, there also exists a developing analytic theory of double product integral. This thesis contributes to this analytic theory. The first papers in that direction are Hudson [2005b] and Hudson and Jones [2012]. Other motivations include quantum extension of Girsanov's theorem and hence a quantum version of the Black-Scholes model in finance. They may also provide a general model for causal interactions in noisy environments in quantum physics. From a different direction "causal" double products, (see Hudson [2005b]), have become of interest in connection with quantum versions of the Levy area, and in particular quantum Levy area formula (Hudson [2011] and Chen and Hudson [2013]) for its characteristic function. There is a close association of causal double products with the double products of rectangular type (Hudson and Jones [2012] pp 3). For this reason it is of interest to study "forwardforward" rectangular double products. In the first chapter we give our notation which will be used in the following chapters and we introduce some simple double products and show heuristically that they are the solution of two different quantum stochastic differential equations. For each example the order in which the products are taken is shown to be unimportant; either calculation gives the same answer. This is in fact a consequence of the so called multiplicative Fubini Theorem Hudson and Pulmaanova [2005]. In Chapter two we formally introduce the notion of product integral as a solution of two particular quantum stochastic differential equations. In Chapter three we introduce the Fock representation of the canonical commutation relations, and discuss the Stone-von Neumann uniqueness theorem. We define the notion of Bogolubov transformation (often called a symplectic automorphism, see Parthasarathy [1992] for example), implementation of these transformations by an implementor (a unitary operator) and introduce Shale's theorem which will be relevant to the following chapters. For an alternative coverage of Shale's Theorem, symplectic automorphism and their implementors see Derezinski [2003]. In Chapter four we study double product integrals of the pseudo rotational type. This is in contrast to double product integrals of the rotational type that have been studied in (Hudson and Jones [2012] and Hudson [2005b]). The notation of the product integral is suggestive of a natural discretisation scheme where the infinitesimals are replaced by discrete increments i.e. discretised creation and annihilation operators of quantum mechanics. Because of a weak commutativity condition, between the discretised creation and annihilation operators corresponding on different subintervals of R, the order of the factors of the product are unimportant (Hudson [2005a]), and hence the discrete product is well defined; we call this result the discrete multiplicative Fubini Theorem. It is also the case that the order in which the products are taken in the continuous (non-discretised case) does not matter (Hudson [2005a], Hudson and Jones [2012]). The resulting discrete double product is shown to be the implementor (a unitary operator) of a Bogolubov transformation acting on discretised creation and annihilation operators (Bogolubov transformations are invertible real linear operators on a Hilbert space that preserve the imaginary part of the inner product, but here we may regard them equivalently as liner transformations acting directly on creation and annihilations operators but preserving adjointness and commutation relations). Unitary operators on the same Hilbert space are a subgroup of the group of Bogolubov transformations. Essentially Bogolubov transformations are used to construct new canonical pairs from old ones (In the literature Bogolubov transformations are often called symplectic automorphisms). The aforementioned Bogolubov transformation (acting on the discretised creation and annihilation operators) can be embedded into the space L2(R+) L2(R+) and limits can be taken resulting in a limiting Bogolubov transformation in the space L2(R+) L2(R+). It has also been shown that the resulting family of Bogolubov transformation has three important properties, namely bi-evolution, shift covariance and time-reversal covariance, see (Hudson [2007]) for a detailed description of these properties. Subsequently we show rigorously that this transformation really is a Bogolubov transformation. We remark that these transformations are Hilbert-Schmidt perturbations of the identity map and satisfy a criterion specified by Shale's theorem. By Shale's theorem we then know that each Bogolubov transformation is implemented in the Fock representation of the CCR. We also compute the constituent kernels of the integral operators making up the Hilbert-Schmidt operators involved in the Bogolubov transformations, and show that the order in which the approximating discrete products are taken has no bearing on the final Bogolubov transformation got by the limiting procedure, as would be expected from the multiplicative Fubini Theorem. In Chapter five we generalise the canonical form of the double product studied in Chapter four by the use of gauge transformations. We show that all the theory of Chapter four carries over to these generalised double product integrals. This is because there is unitary equivalence between the Bogolubov transformation got from the generalised canonical form of the double product and the corresponding original one. In Chapter six we make progress towards showing that a system of implementors of this family of Bogolubov transformations can be found which inherits properties of the original family such as being a bi-evolution and being covariant under shifts. We make use of Shales theorem (Parthasarathy [1992] and Derezinski [2003]). More specifically, Shale's theorem ensures that each Bogolubov transformation of our system is implemented by a unitary operator which is unique to with multiplicaiton by a scalar of modulus 1. We expect that there is a unique system of implementors, which is a bi-evolution, shift covariant, and time reversal covariant (i.e. which inherits the properties of the corresponding system of Bogolubov transformation). This is partly on-going research. We also expect the implementor of the Bogolubov transformation to be the original double product. In Evans [1988], Evan's showed that the the implementor of a Bogolubov transformation in the simple product case is indeed the simple product. If given more time it might be possible to adapt Evan's result to the double product case.

The probit function is the inverse of the cumulative distribution function associated with the standard normal distribution. It is of great utility in statistical modelling. The Carleman embedding technique has been shown to be effective in solving first order and, less efficiently, second order nonlinear differential equations. In this thesis, we show that solutions to the second order nonlinear differential equation for the probit function can be approximated efficiently using a variant of the Carleman embedding technique.

Ce travail étend nos connaissances des entropies libres et des équations différentielles stochastiques (EDS) libres dans trois directions. Dans un premier temps, nous montrons que l'algèbre de von Neumann engendrée par au moins deux autoadjoints ayant une information de Fisher finie n'a pas la propriété $Gamma$ de Murray et von Neumann. C'est un analogue d'un résultat de Voiculescu pour l'entropie microcanonique libre. Dans un second temps, nous étudions des EDS libres à coefficients opérateurs non-bornés (autrement dit des sortes d' EDP stochastiques libres ). Nous montrons la stationnarité des solutions dans des cas particuliers. Nous en déduisons un calcul de la dimension entropique libre microcanonique dans le cas d'une information de Fisher lipschitzienne. Dans un troisième et dernier temps, nous introduisons une méthode générale de résolutions d'EDS libres stationnaires, s'appuyant sur un analogue non-commutatif d'un espace de chemins. En définissant des états traciaux sur cet analogue, nous construisons des dilatations markoviennes de nombreux semigroupes complètement markoviens sur une algèbre de von Neumann finie, en particulier de tous les semigroupes symétriques. Pour des semigroupes particuliers, par exemple dès que le générateur s'écrit sous une forme divergence pour une dérivation à valeur dans la correspondance grossière, ces dilatations résolvent des EDS libres. Entre autres applications, nous en déduisons une inégalité de Talagrand pour l'entropie non-microcanonique libre (relative à une sous-algèbre et une application complètement positive). Nous utilisons aussi ces déformations dans le cadre des techniques de déformations/rigidité de Popa
This works extends our knowledge of free entropies, free Fisher information and free stochastic differential equations in three directions. First, we prove that if a $W^{*}$-probability space generated by more than 2 self-adjoints with finite non-microstates free Fisher information doesn't have property $Gamma$ of Murray and von Neumann (especially is not amenable). This is an analogue of a well-known result of Voiculescu for microstates free entropy. We also prove factoriality under finite non-microstates entropy. Second, we study a general free stochastic differential equation with unbounded coefficients (``stochastic PDE"), and prove stationarity of solutions in well-chosen cases. This leads to a computation of microstates free entropy dimension in case of Lipschitz conjugate variable. Finally, we introduce a non-commutative path space approach to solve general stationary free Stochastic differential equations. By defining tracial states on a non-commutative analogue of a path space, we construct Markov dilations for a class of conservative completely Markov semigroups on finite von Neumann algebras. This class includes all symmetric semigroups. For well chosen semigroups (for instance with generator any divergence form operator associated to a derivation valued in the coarse correspondence) those dilations give rise to stationary solutions of certain free SDEs. Among applications, we prove a non-commutative Talagrand inequality for non-microstate free entropy (relative to a subalgebra $B$ and a completely positive map $eta:Bto B$). We also use those new deformations in conjunction with Popa's deformation/rigidity techniques, to get absence of Cartan subalgebra results

A stochastic differential equation with vanishing martingale term is studied. Specifically, given a domain D, the asymptotic scaling properties of both the exit time from the domain and the exit distribution are considered under the additional (non-standard) hypothesis that the initial condition also has a scaling limit. Methods from dynamical systems are applied to get more complete estimates than the ones obtained by the probabilistic large deviation theory. Two situations are completely analyzed. When there is a unique critical saddle point of the deterministic system (the system without random effects), and when the unperturbed system escapes the domain D in finite time. Applications to these results are in order. In particular, the study of 2-dimensional heteroclinic networks is closed with these results and shows the existence of possible asymmetries. Also, 1-dimensional diffusions conditioned to rare events are further studied using these results as building blocks. The approach tries to mimic the well known linear situation. The original equation is smoothly transformed into a very specific non-linear equation that is treated as a singular perturbation of the original equation. The transformation provides a classification to all 2-dimensional systems with initial conditions close to a saddle point of the flow generated by the drift vector field. The proof then proceeds by estimates that propagate the small noise nature of the system through the non-linearity. Some proofs are based on geometrical arguments and stochastic pathwise expansions in noise intensity series.

By introducing a concept of dynamic process operating under multi-time scales in sciences and engineering, a mathematical model is formulated and it leads to a system of multi-time scale stochastic differential equations. The classical Picard-Lindel\"{o}f successive approximations scheme is expended to the model validation problem, namely, existence and uniqueness of solution process. Naturally, this generates to a problem of finding closed form solutions of both linear and nonlinear multi-time scale stochastic differential equations. To illustrate the scope of ideas and presented results, multi-time scale stochastic models for ecological and epidemiological processes in population dynamic are exhibited. Without loss in generality, the modeling and analysis of three time-scale fractional stochastic differential equations is followed by the development of the numerical algorithm for multi-time scale dynamic equations. The development of numerical algorithm is based on the idea if numerical integration in the context of the notion of multi-time scale integration. The multi-time scale approach is applied to explore the study of higher order stochastic differential equations (HOSDE) is presented. This study utilizes the variation of constant parameter technique to develop a method for finding closed form solution processes of classes of HOSDE. Then then probability distribution of the solution processes in the context of the second order equations is investigated.

The increasing importance of data in the modern world has created a need for new mathematical techniques to analyze this data. We explore and develop the use of geometry—specifically differential geometry—as a means for such analysis, in two parts. First, we provide a general framework to discover patterns contained in time series data using a geometric framework of assigning distance, clustering, and then forecasting. Second, we attempt to define a Riemannian metric on the space containing the data in order to introduce a notion of distance intrinsic to the data, providing a novel way to probe the data for insight.

In this thesis, we consider a class of stochastic dynamics running backwards, so called backward stochastic differential equations (BSDEs) in the literature. We demonstrate BSDEs can be reformulated as functional differential equations defined on path spaces, and therefore solving BSDEs is equivalent to solving the associated functional differential equations. With such observation we can solve BSDEs on general filtered probability space satisfying the usual conditions, and in particular without the requirement of the martingale representation. We further solve the above functional differential equations numerically, and propose a numerical scheme based on the time discretization and the Picard iteration. This in turn also helps us solve the associated BSDEs numerically. In the second part of the thesis, we consider a class of BSDEs with quadratic growth (QBSDEs). By using the functional differential equation approach introduced in this thesis and the idea of the Cole-Hopf transformation, we first solve the scalar case of such QBSDEs on general filtered probability space satisfying the usual conditions. For a special class of QBSDE systems (not necessarily scalar) in Brownian setting, we do not use such Cole-Hopf transformation at all, and instead introduce the weak solution method, which is to use the strong solutions of forward backward stochastic differential equations (FBSDEs) to construct the weak solutions of such QBSDE systems. Finally we apply the weak solution method to a specific financial problem in the credit risk setting, where we modify the Merton's structural model for credit risk by using the idea of indifference pricing. The valuation and the hedging strategy are characterized by a class of QBSDEs, which we solve by the weak solution method.

Auditory physiology is nearly unique in the human body because of its small-diameter neurons. When considering a single node on one neuron, the number of channels is very small, so ion fluxes exhibit randomness. Hodgkin and Huxley, in 1952, set forth a system of Ordinary Differential Equations (ODEs) to track the flow of ions in a squid motor neuron, based on a circuit analogy for electric current. This formalism for modeling is still in use today and is useful because coefficients can be directly measured. To measure auditory properties of Firing Efficiency (FE) and Post Stimulus Time (PST), we can simply measure the depolarization, or "upstroke," of a node. Hence, we reduce the four-dimensional squid neuron model to a two-dimensional system of ODEs. The stochastic variable m for sodium activation is allowed a random walk in addition to its normal evolution, and the results are drastic. The diffusion coefficient, for spreading, is inversely proportional to the number of channels; for 130 ion channels, D is closer to 1/3 than 0 and cannot be called negligible. A system of Partial Differential Equations (PDEs) is derived in these pages to model the distribution of states of the node with respect to the (nondimensionalized) voltage v and the sodium activation gate m. Initial conditions describe a distribution of (v,m) states; in most experiments, this would be a curve with mode at the resting state. Boundary conditions are Robin (Natural) boundary conditions, which gives conservation of the population. Evolution of the PDE has a drift term for the mean change of state and a diffusion term, the random change of state. The phase plane is broken into fired and resting regions, which form basins of attraction for fired and resting-state fixed points. If a stimulus causes ions to flow from the resting region into the fired region, this rate of flux is approximately the firing rate, analogous to clinically measuring when the voltage crosses a threshold. This gives a PST histogram. The FE is an integral of the population over the fired region at a measured stop time after the stimulus (since, in the reduced model, when neurons fire they do not repolarize). This dissertation also includes useful generalizations and methodology for turning other ODEs into PDEs. Within the HH modeling, parameters can be switched for other systems of the body, and may present a similar firing and non-firing separatrix (as in Chapter 3). For any system of ODEs, an advection model can show a distribution of initial conditions or the evolution of a given initial probability density over a state space (Chapter 4); a system of Stochastic Differential Equations can be modeled with an advection-diffusion equation (Chapter 5). As computers increase in speed and as the ability of software to create adaptive meshes and step sizes improves, modeling with a PDE becomes more and more efficient over its ODE counterpart.

This thesis is concerned with mathematical modelling of oncolytic virotherapy: the use of genetically modified viruses to selectively spread, replicate and destroy cancerous cells in solid tumours. Traditional spatially-dependent modelling approaches have previously assumed that virus spread is due to viral diffusion in solid tumours, and also neglect the time delay introduced by the lytic cycle for viral replication within host cells. A deterministic, age-structured reaction-diffusion model is developed for the spatially-dependent interactions of uninfected cells, infected cells and virus particles, with the spread of virus particles facilitated by infected cell motility and delay. Evidence of travelling wave behaviour is shown, and an asymptotic approximation for the wave speed is derived as a function of key parameters. Next, the same physical assumptions as in the continuum model are used to develop an equivalent discrete, probabilistic model for that is valid in the limit of low particle concentrations. This mesoscopic, compartment-based model is then validated against known test cases, and it is shown that the localised nature of infected cell bursts leads to inconsistencies between the discrete and continuum models. The qualitative behaviour of this stochastic model is then analysed for a range of key experimentally-controllable parameters. Two-dimensional simulations of in vivo and in vitro therapies are then analysed to determine the effects of virus burst size, length of lytic cycle, infected cell motility, and initial viral distribution on the wave speed, consistency of results and overall success of therapy. Finally, the experimental difficulty of measuring the effective motility of cells is addressed by considering effective medium approximations of diffusion through heterogeneous tumours. Considering an idealised tumour consisting of periodic obstacles in free space, a two-scale homogenisation technique is used to show the effects of obstacle shape on the effective diffusivity. A novel method for calculating the effective continuum behaviour of random walks on lattices is then developed for the limiting case where microscopic interactions are discrete.

This thesis is concerned with the mathematical analysis of electricity and carbon emission markets. We introduce a novel, versatile and tractable stochastic framework for the joint price formation of electricity spot prices and allowance certificates. In the proposed framework electricity and allowance prices are explained as functions of specific fundamental factors, such as the demand for electricity and the prices of the fuels used for its production. As a result, the proposed model very clearly captures the complex dependency of the modelled prices on the aforementioned fundamental factors. The allowance price is obtained as the solution to a coupled forward-backward stochastic differential equation. We provide a rigorous proof of the existence and uniqueness of a solution to this equation and analyse its behaviour using asymptotic techniques. The essence of the model for the electricity price is a carefully chosen and explicitly constructed function representing the supply curve in the electricity market. The model we propose accommodates most regulatory features that are commonly found in implementations of emissions trading systems and we analyse in detail the impact these features have on the prices of allowance certificates. Thereby we reveal a weakness in existing regulatory frameworks, which, in rare cases, can lead to allowance prices that do not conform with the conditions imposed by the regulator. We illustrate the applicability of our model to the pricing of derivative contracts, in particular clean spread options and numerically illustrate its ability to "see" relationships between the fundamental variables and the option contract, which are usually unobserved by other commonly used models in the literature. The results we obtain constitute flexible tools that help to efficiently evaluate the financial impact current or future implementations of emissions trading systems have on participants in these markets.

Cardiovascular disease is the primary cause of death globally. Although this group encompasses a heterogeneous range of conditions, many of these diseases are associated with abnormalities in the cardiac electrical propagation. In these conditions, structural abnormalities in the form of scars and fibrotic tissue are known to play an important role, leading to a high individual variability in the exact disease mechanisms. Because of this, clinical interventions such as ablation therapy and CRT that work by modifying the electrical propagation should ideally be optimized on a patient specific basis. As a tool for optimizing these interventions, computational modelling and simulation of the heart have become increasingly important. However, in order to construct these models, a crucial step is the estimation of tissue conduction properties, which have a profound impact on the cardiac activation sequence predicted by simulations. Information about the conduction properties of the cardiac tissue can be gained from electrophysiological data, obtained using electroanatomical mapping systems. However, as in other clinical modalities, electrophysiological data are often sparse and noisy, and this results in high levels of uncertainty in the estimated quantities. In this dissertation, we develop a methodology based on Bayesian inference, together with a computationally efficient model of electrical propagation to achieve two main aims: 1) to quantify values and associated uncertainty for different tissue conduction properties inferred from electroanatomical data, and 2) to design strategies to optimise the location and number of measurements required to maximise information and reduce uncertainty. The methodology is validated in several studies performed using simulated data obtained from image-based ventricular models, including realistic fibre orientation and conduction heterogeneities. Subsequently, by using the developed methodology to investigate how the uncertainty decreases in response to added measurements, we derive an a priori index for placing electrophysiological measurements in order to optimise the information content of the collected data. Results show that the derived index has a clear benefit in minimising the uncertainty of inferred conduction properties compared to a random distribution of measurements, suggesting that the methodology presented in this dissertation provides an important step towards improving the quality of the spatiotemporal information obtained using electroanatomical mapping.

There exist many numerical methods for numerical solutions of the systems of stochastic differential equations. We choose the method of deterministic quadrature formulae proposed by Müller–Gronbach, and Yaroslavtseva in 2016. The idea is to apply a simplified version of the cubature in Wiener space. We explain the method and check how good it works in the simplest case of the classical Black–Scholes model.

Monte Carlo path simulations are common in mathematical and computational finance as a way of estimating the expected values of a quantity such as a European put option, which is functional to the solution of a stochastic differential equation (SDE). The computational complexity of the standard Monte Carlo (MC) method grows quite large quickly, so in this thesis we focus on the Multilevel Monte Carlo (MLMC) method by Giles, which uses multigrid ideas to reduce the computational complexity. We use a Euler-Maruyama time discretisation for the approximation of the SDE and investigate how the convergence rate of the MLMC method improves the computational times and cost in comparison with the standard MC method. We perform a numerical analysis on the computational times and costs in order to achieve the desired accuracy and present our findings on the performance of the MLMC method on a European put option compared to the standard MC method.

La these comporte deux parties independantes. La premiere partie traite de l'utilisation de la geometrie differentielle en statistique. Dans les chapitres i et ii sont rappelees les notions fondamentales de geometrie differentielle et de statistique. Le chapitre iii est consacre a la theorie des "strings" dont de nombreux exemples apparaissent en statistique inferentielle. Ce nouveau concept a ete introduit et etudie par m. M. O. E. Barndorff nielsen et p. Blaesild. Nous en donnons ici une nouvelle definition, purement mathematique, basee sur un concept de differenciation d'ordre superieur, les objets differencies etant des fonctions, champs de vecteurs tangents, contangents ou jets. Enfin, dans le chapitre iv, a partir de structures geometriques specifiques definies sur des modeles statistiques parametriques reguliers et basees sur un point de vue de conditionnement pour une statistique ancillaire donnee, nous elaborons des developpements asymptotiques pour les lois du vecteur score et de l'estimateur du maximum de vraisemblance. La seconde partie concerne les familles exponentielles naturelles k-dimensionnelles et les fonctions-variance qui les caracterisent. Dans ce contexte nous etablissons dans le chapitre v un theoreme de convergence qui montre que l'ensemble des fonctions variances est ferme pour la convergence uniforme sur tout compact

Stochastic dynamic programming is a recursive method for solving sequential or multistage decision problems. It helps economists and mathematicians construct and solve a huge variety of sequential decision making problems in stochastic cases. Research on stochastic dynamic programming is important and meaningful because stochastic dynamic programming reflects the behavior of the decision maker without risk aversion; i.e., decision making under uncertainty. In the solution process, it is extremely difficult to represent the existing or future state precisely since uncertainty is a state of having limited knowledge. Indeed, compared to the deterministic case, which is decision making under certainty, the stochastic case is more realistic and gives more accurate results because the majority of problems in reality inevitably have many unknown parameters. In addition, time scale calculus theory is applicable to any field in which a dynamic process can be described with discrete or continuous models. Many stochastic dynamic models are discrete or continuous, so the results of time scale calculus are directly applicable to them as well. The aim of this thesis is to introduce a general form of a stochastic dynamic sequence problem on complex discrete time domains and to find the optimal sequence which maximizes the sequence problem.

In this paper, we consider the capture dynamics of a particle undergoing a random walk above a sheet of absorbing traps. In particular, we seek to characterize the distribution in time from when the particle is released to when it is absorbed. This problem is motivated by the study of lymphocytes in the human blood stream; for a particle near the surface of a lymphocyte, how long will it take for the particle to be captured? We model this problem as a diffusive process with a mixture of reflecting and absorbing boundary conditions. The model is analyzed from two approaches. The first is a numerical simulation using a Kinetic Monte Carlo (KMC) method that exploits exact solutions to accelerate a particle-based simulation of the capture time. A notable advantage of KMC is that run time is independent of how far from the traps one begins. We compare our results to the second approach, which is asymptotic approximations of the FPT distribution for particles that start far from the traps. Our goal is to validate the efficacy of homogenizing the surface boundary conditions, replacing the reflecting (Neumann) and absorbing (Dirichlet) boundary conditions with a mixed (Robin) boundary condition.

Дипломну роботу виконано на 43 аркушах, вона містить 2 додатки та перелік посилань на використані джерела з 16 найменувань. У роботі наведено 3 рисунки та 4 таблиці. Метою даної дипломної роботи є аналiз, уточнення та застосування м є т о д ів дослідження марковських SP-мереж на с т ій к і с т ь до диференціального криптоаналізу. Обєктом дослідження є інформаційні процеси в системах криптографічного захисту. Предметом дослідження є алгоритми оцінювання SP-мереж на стійкість до диференціального криптоаналізу. В роботі проводиться уточнення та застосування методів для оцінки стійкості SP-мереж до диференціального криптоаналізу на прикладі шифру ДСТУ 7624:2014. Основні положення дипломної роботи опубліковано у вигляді тез доповіді на Міжнародній науково-практичній конференції БШ!ТС 2016 та Всеукраїнській науково-практичній коференції ТШПФМТ! 2016.
The thesis is presented in 43 pages. It contains 2 appendixes and bibliography of 16 references. Four figures and 2 tables are given in the thesis. The goal of the thesis is the analysis, specification and application of research methods Markov SP-networks for resistance to differential cryptanalysis. The object of research is the information processes in cryptographic protection systems. The subject of research is estimation of SP-networks algorithms for resistance to differential cryptanalysis. In the presented thesis the Markov SP-networks resistance to differential cryptoanalysis is assessed, taking the DSTU 7624:2014 cipher as the illustrative example. Main ideas of the thesis were published in the Proceedings of the International Practical and Technical Conference БГО!ТС 2016.
Дипломную работу выполнено на 43 листах, она содержит 2 приложения и перечень использованных источников из 16 наименований. В работе приведены 3 рисунки и 4 таблицы. Целью данной работы является анализ, уточнение и применение методов исследования марковских SP-сетей на устойчивость к дифференциальному криптоанализу. Объектом исследования являются процессы в системах криптографической защиты. Предметом исследования являются алгоритмы оценки SP-сетей на устойчивость к дифференциальному криптоанализу. В работе проводится уточнение и применение методов для оценки устойчивости SP-сетей к дифференциальному криптоанализу на примере шифра ДСТУ 7624:2014. Основные положения дипломной работы опубликованы в виде тезисов на Международной научно-практической конференции БИВИТС 2016 и Всеукраинской научно-практической коференции ТИППФМТИ 2016.

Dans cette thèse, je me suis interessé a deux aspects de la gestion de portefeuille : la maximisation de l'utilité e d'un portefeuille financier lorsque on impose une contrainte sur l'exposition au risque, et la couverture quadratique en marché incomplet. Part I. Dans la première partie, j' étudie un problème d'assurance de portefeuille du point de vue du manager d'un fond d'investissement, qui veut structurer un produit financier pour les investisseurs du fond avec une garantie sur la valeur du portefeuille a la maturité . Si, a la maturité, la valeur du portefeuille est au dessous d'un seuil x e, l'investisseur sera remboursé a la hauteur de ce seuil par une troisième partie, qui joue le rôle d'assureur du fond (on peut imaginer que le fond appartient à une banque et que donc c'est la banque elle même qui joue le rôle d'assureur). En échange de cette assurance, la troisième partie impose une contrainte sur l'exposition au risque que le manager du fond peut tolérer, mesurée avec une mesure de risque monétaire convexe. Je donne la solution complet e de ce problème de maximisation non convexe en marché complet et je prouve que le choix de la mesure de risque est un point crucial pour avoir existence d'un portefeuille optimal. J'applique donc mes résultats lorsque on utilise la mesure de risque entropique (pour laquelle le portefeuille optimal existe toujours), les mesures de risque spectrales (pour lesquelles le portefeuille optimal peut ne pas exister dans certains cas) et la G-divergence. Mots-cl es : Assurance de portefeuille ; maximisation d'utilité ; mesure de risque convexe ; VaR, CVaR et mesure de risque spectrale ; entropie et G-divergence. Part II. Dans la deuxième partie, je m'intéresse au problème de couverture quadratique en marché incomplet. J'assume que le marché est d écrit par un processus Markovien tridimensionnel avec sauts. La premi ère variable d' état décrit l'actif - financier, échangeable sur le marché, qui sert comme instrument de couverture ; la deuxième variable d' état modélise un actif financier que intervient dans la dynamique de l'instrument de couverture mais qui n'est pas échangeable sur le march é : il peut donc être vu comme un facteur de volatilité de l'instrument de couverture, ou comme un actif financier que l'on ne peut pas acheter (pour de raisons légales par exemple) ; la troisième et dernière variable d' état représente une source externe de risque qui affecte l'option Européenne qu'on veut couvrir, et qui, elle aussi, n'est pas échangeable sur le marché. Pour résoudre le problème j'utilise l'approche de la programmation dynamique, qui me permet d' écrire l' équation de Hamilton-Jacobi- Bellman associé e au problème de couverture quadratique, qui est non locale en non linéaire. Je prouve que la fonction valeur associée au problème de couverture quadratique peut être caractérisée par un système de trois équations integro- différentielles aux dérivées partielles, dont l'une est semilinéaire et ne dépends pas du choix de l'option a couvrir, et les deux autres sont simplement linéaires , et que ce système a une unique solution r régulière dans un espace de Hölder approprié, qui me permet donc de caractériser la stratégie de couverture optimale . Ce résultat est démontré lorsque le processus est non dégénéré (c'est a dire que la composante Brownienne est strictement elliptique) et lorsque le processus est a sauts purs. Je conclus avec une application de mes résultats dans le cadre du marché de l' électricité. Mots-cl es : Couverture quadratique ; modèle a sauts ; programmation dynamique ; équation de Hamilton-Jacobi-Bellman ; équations aux dérivées partielles integro-différentielles.

Statistical analysis and modeling are useful for understanding the behavior of different phenomena. In this study we will focus on two areas of applications: Global warming and cancer research. Global Warming is one of the major environmental challenge people face nowadays and cancer is one of the major health problem that people need to solve. For Global Warming, we are interest to do research on two major contributable variables: Carbon dioxide (CO2) and atmosphere temperature. We will model carbon dioxide in the atmosphere data with a system of differential equations. We will develop a differential equation for each of six attributable variables that constitute CO2 in the atmosphere and a differential system of CO2 in the atmosphere. We are using real historical data on the subject phenomenon to develop the analytical form of the equations. We will evaluate the quality of the developed model by utilizing a retrofitting process. Having such an analytical system, we can obtain good estimates of the rate of change of CO2 in the atmosphere, individually and cumulatively as a function of time for near and far target times. Such information is quite useful in strategic planning of the subject matter. We will develop a statistical model taking into consideration all the attributable variables that have been identified and their corresponding response of the amount of CO2 in the atmosphere in the continental United States. The development of the statistical model that includes interactions and higher order entities, in addition to individual contributions to CO2 in the atmosphere, are included in the present study. The proposed model has been statistically evaluated and produces accurate predictions for a given set of the attributable variables. Furthermore, we rank the attributable variables with respect to their significant contribution to CO2 in the atmosphere. For Cancer Research, the object of the study is to probabilistically evaluate commonly used methods to perform survival analysis of medical patients. Our study includes evaluation of parametric, semi-parametric and nonparametric analysis of probability survival models. We will evaluate the popular Kaplan-Meier (KM), the Cox Proportional Hazard (Cox PH), and Kernel density (KD) models using both Monte Carlo simulation and using actual breast cancer data. The first part of the evaluation will be based on how these methods measure up to parametric analysis and the second part using actual cancer data. As expected, the parametric survival analysis when applicable gives the best results followed by the not commonly used nonparametric Kernel density approach for both evaluations using simulation and actual cancer data. We will develop a statistical model for breast cancer tumor size prediction for United States patients based on real uncensored data. When we simulate breast cancer tumor size, most of time these tumor sizes are randomly generated. We want to construct a statistical model to generate these tumor sizes as close as possible to the real patients' data given other related information. We accomplish the objective by developing a high quality statistical model that identifies the significant attributable variables and interactions. We rank these contributing entities according to their percentage contribution to breast cancer tumor growth. This proposed statistical model can also be used to conduct surface response analysis to identify the necessary restrictions on the significant attributable variables and their interactions to minimize the size of the breast tumor. We will utilize the Power Law process, also known as Non-homogenous Poisson Process and Weibull Process to evaluate the effectiveness of a given treatment for Stage I & II Ductal breast cancer patients. We utilize the shape parameter of the intensity function to evaluate the behavior of a given treatment with respect to its effectiveness. We will develop a differential equation that will characterize the behavior of the tumor as a function of time. Having such a differential equation, the solution of which once plotted will identify the rate of change of tumor size as a function of age. The structure of the differential equation consists of the significant attributable variables and their interactions to the growth of breast cancer tumor. Once we have developed the differential equations and its solution, we proceed to validate the quality of the proposed differential equations and its usefulness.

Dropped object are defined as any object that fall under its own weight from a previously static position or fell due to an applied force from equipment or a moving object. It is among the top ten causes of injuries and fatality in oil and gas industry. To solve this problem, several in-house tools and guidelines is developed over time to assess the risk of dropped objects on the sub-sea structures. This thesis focuses on compiling and comparing those methods in hope to improve the recommended practices available in the market. A simple modification is done on the in-house tools to better predict the landing point distribution of the dropped cylindrical objects on the seabed by imposing the random three-dimensional rotation around the water depth axis. This tool is then used to compare the result of annual hit frequency using the recommended practice and further compared with the available experimental data.

Thesis (M.S. in Operations Research and M.S. in Applied Mathematics)--Naval Postgraduate School, June 2004.
Thesis advisor(s): Craig W. Rasmussen, Thomas M. Cioppa. Includes bibliographical references (p. 131-132). Also available online.

In this thesis, the structural model in credit risk and the credit derivatives is studied under both Black-Scholes setting and Variance Gamma (VG) setting. Using a Variance Gamma process, the distribution of the firm value process becomes asymmetric and leptokurtic. Also, the jump structure of VG processes allows random default times of the reference entities. Among structural models, the most emphasis is made on the Black-Cox model by building a relation between the survival probabilities of the Black-Cox model and the value of a binary down and out barrier option. The survival probabilities under VG setting are calculated via a Partial Integro Differential Equation (PIDE). Some applications of binary down and out barrier options, default probabilities and Credit Default Swap par spreads are also illustrated in this study.

Stochastic models describing how interacting individuals give rise to collective behaviour have become a widely used tool across disciplines—ranging from biology to physics to social sciences. Continuum population-level models based on partial differential equations for the population density can be a very useful tool (when, for large systems, particle-based models become computationally intractable), but the challenge is to predict the correct macroscopic description of the key attributes at the particle level (such as interactions between individuals and evolution rules). In this thesis we consider the simple class of models consisting of diffusive particles with short-range interactions. It is relevant to many applications, such as colloidal systems and granular gases, and also for more complex systems such as diffusion through ion channels, biological cell populations and animal swarms. To derive the macroscopic model of such systems, previous studies have used ad hoc closure approximations, often generating errors. Instead, we provide a new systematic method based on matched asymptotic expansions to establish the link between the individual- and the population-level models. We begin by deriving the population-level model of a system of identical Brownian hard spheres. The result is a nonlinear diffusion equation for the one-particle density function with excluded-volume effects enhancing the overall collective diffusion rate. We then expand this core problem in several directions. First, for a system with two types of particles (two species) we obtain a nonlinear cross-diffusion model. This model captures both alternative notions of diffusion, the collective diffusion and the self-diffusion, and can be used to study diffusion through obstacles. Second, we study the diffusion of finite-size particles through confined domains such as a narrow channel or a Hele–Shaw cell. In this case the macroscopic model depends on a confinement parameter and interpolates between severe confinement (e.g., a single- file diffusion in the narrow channel case) and an unconfined situation. Finally, the analysis for diffusive soft spheres, particles with soft-core repulsive potentials, yields an interaction-dependent non-linear term in the diffusion equation.

The goal of this thesis in finance is to combine the use of advanced mathematical methods with a return to foundational economic issues. In that perspective, I study generalized rational expectations and asset pricing in Chapter 2, and a converse comparison principle for backward stochastic differential equations with jumps in Chapter 3. Since the use of stochastic methods in finance is an interesting and complex issue in itself - if only to clarify the difference between the use of mathematical models in finance and in physics or biology - I also present a philosophical reflection on the interpretation of mathematical models in finance (Chapter 4). In Chapter 5, I conclude the thesis with an essay on the history and interpretation of mathematical probability - to be read while keeping in mind the fundamental role of mathematical probability in financial models.
Doctorat en Sciences économiques et de gestion
info:eu-repo/semantics/nonPublished

Cette thèse a pour objet la simulation d'une zone de mélange turbulente issue de l'instabilité de Richtmyer-Meshkov à l'aide d'un modèle à fonction de densité de probabilité (PDF). Nous analysons plus particulièrement la prise en charge par le modèle PDF du transport de l'énergie cinétique turbulente dans la zone de mélange.Dans cette optique, nous commençons par mettre en avant le lien existant entre les statistiques en un point de l'écoulement et ses conditions initiales aux grandes échelles. Ce lien s'exprime à travers le principe de permanence des grandes échelles, et permet d'établir des prédictions pour certaines grandeurs de la zone de mélange, telles que son taux de croissance ou son anisotropie.Nous dérivons ensuite un modèle PDF de Langevin capable de restituer cette dépendance aux conditions initiales. Ce modèle est ensuite validé en le comparant à des résultats issus de simulations aux grandes échelles (LES).Enfin, une analyse asymptotique du modèle proposé permet d'éclairer notre compréhension du transport turbulent. Un régime de diffusion est mis en évidence, et l'expression du coefficient de diffusion associé à ce régime atteste l'influence de la permanence des grandes échelles sur le transport turbulent.Tout au long de cette thèse, nous nous sommes appuyés sur des résultats issus de simulations de Monte Carlo du modèle de Langevin. A cet effet, nous avons développé une méthode spécifique eulérienne et à l'avons comparé à des alternatives lagrangiennes
The aim of the thesis is to simulate a turbulent mixing zone resulting from the Richtmyer-Meshkov instability using a probability density function (PDF) model. An emphasis is put on the analysis of the turbulent kinetic energy transport.To this end, we first highlight the link existing between the one-point statistics of the flow and its initial conditions at large scales. This link is expressed through the principle of permanence of large eddies, and allows to establish predictions for quantities of the mixing zone, such as its growth rate or its anisotropy.We then derive a Langevin PDF model which is able to reproduce this dependency of the statistics on the initial conditions. This model is then validated by comparing it against large eddy simulations (LES).Finally, an asymptotic analysis of the derived model helps to improve our understanding of the turbulent transport. A diffusion regime is identified, and the expression of the diffusion coefficient associated with this regime confirms the influence of the permanence of large eddies on the turbulent transport.Throughout this thesis, our numerical results were based on Monte Carlo simulations for the Langevin model. In this regard, we proceeded to the development of a specific Eulerian method and its comparison with Lagrangian counterparts

Dans cette thèse, nous proposons une approche progressive (forward) pour la représentation probabiliste d'Equations aux Dérivées Partielles (EDP) nonlinéaires et nonconservatives, permettant ainsi de développer un algorithme particulaire afin d'en estimer numériquement les solutions. Les Equations Différentielles Stochastiques Nonlinéaires de type McKean (NLSDE) étudiées dans la littérature constituent une formulation microscopique d'un phénomène modélisé macroscopiquement par une EDP conservative. Une solution d'une telle NLSDE est la donnée d'un couple $(Y,u)$ où $Y$ est une solution d' équation différentielle stochastique (EDS) dont les coefficients dépendent de $u$ et de $t$ telle que $u(t,cdot)$ est la densité de $Y_t$. La principale contribution de cette thèse est de considérer des EDP nonconservatives, c'est-à- dire des EDP conservatives perturbées par un terme nonlinéaire de la forme $Lambda(u,nabla u)u$. Ceci implique qu'un couple $(Y,u)$ sera solution de la représentation probabiliste associée si $Y$ est un encore un processus stochastique et la relation entre $Y$ et la fonction $u$ sera alors plus complexe. Etant donnée la loi de $Y$, l'existence et l'unicité de $u$ sont démontrées par un argument de type point fixe via une formulation originale de type Feynmann-Kac
This thesis performs forward probabilistic representations of nonlinear and nonconservative Partial Differential Equations (PDEs), which allowto numerically estimate the corresponding solutions via an interacting particle system algorithm, mixing Monte-Carlo methods and non-parametric density estimates.In the literature, McKean typeNonlinear Stochastic Differential Equations (NLSDEs) constitute the microscopic modelof a class of PDEs which are conservative. The solution of a NLSDEis generally a couple $(Y,u)$ where $Y$ is a stochastic process solving a stochastic differential equation whose coefficients depend on $u$ and at each time $t$, $u(t,cdot)$ is the law density of the random variable $Y_t$.The main idea of this thesis is to consider this time a non-conservative PDE which is the result of a conservative PDE perturbed by a term of the type $Lambda(u, nabla u) u$. In this case, the solution of the corresponding NLSDE is again a couple $(Y,u)$, where again $Y$ is a stochastic processbut where the link between the function $u$ and $Y$ is more complicated and once fixed the law of $Y$, $u$ is determined by a fixed pointargument via an innovating Feynmann-Kac type formula

This thesis is concerned with methodologies for the accurate quantitative modelling of molecular biological systems. The first part is devoted to the chemical Langevin equation (CLE), a stochastic differential equation driven by a multidimensional Wiener process. The CLE is an approximation to the standard discrete Markov jump process model of chemical reaction kinetics. It is valid in the regime where molecular populations are abundant enough to assume their concentrations change continuously, but stochastic fluctuations still play a major role. We observe that the CLE is not a single equation, but a family of equations with shared finite-dimensional distributions. On the theoretical side, we prove that as many Wiener processes are sufficient to formulate the CLE as there are independent variables in the equation, which is just the rank of the stoichiometric matrix. On the practical side, we show that in the case where there are m_1 pairs of reversible reactions and m_2 irreversible reactions, there is another, simple formulation of the CLE with only m_1+m_2 Wiener processes, whereas the standard approach uses 2m_1+m_2. Considerable computational savings are achieved with this latter formulation. A flaw of the CLE model is identified: trajectories may leave the nonnegative orthant with positive probability. The second part addresses the challenge when alternative, structurally different ordinary differential equation models of similar complexity fit the available experimental data equally well. We review optimal experiment design methods for choosing the initial state and structural changes on the biological system to maximally discriminate between the outputs of rival models in terms of L_2-distance. We determine the optimal stimulus (input) profile for externally excitable systems. The numerical implementation relies on sum of squares decompositions and is demonstrated on two rival models of signal processing in starving Dictyostelium amoebae. Such experiments accelerate the perfection of our understanding of biochemical mechanisms.

In under 6 decades dengue has emerged from South East Asia to become the most widespread arbovirus affecting human populations. Recent dramatic increases in epidemic dengue fever have mainly been attributed to factors such as vector expansion and ongoing ecological, climate and socio-demographic changes. The failure to control the virus in endemic regions and prevent global spread of its mosquito vectors and genetic variants, underlines the urgency to reassess previous research methods, hypotheses and empirical observations. This thesis comprises a set of studies that integrate currently neglected and emerging epidemiological, ecological and evolutionary factors into unified mathematical frameworks, in order to better understand the contemporary population biology of the dengue virus. The observed epidemiological dynamics of dengue are believed to be driven by selective forces emerging from within-host cross-immune reactions during sequential, heterologous infections. However, this hypothesis is mainly supported by modelling approaches that presume all hosts to contribute equally and significantly to the selective effects of cross-immunity both in time and space. In the research presented in this thesis it is shown that the previously proposed effects of cross-immunological reactions are weakened in agent-based modelling approaches, which relax the common deterministic and homogeneous mixing assumptions in host-host and host-pathogen interactions. Crucially, it is shown that within these more detailed models, previously reported universal signatures of dengue's epidemiology and population genetics can be reproduced by demographic and natural stochastic processes alone. While this contrasts with the proposed role of cross-immunity, it presents demographic stochasticity as a parsimonious mechanism that integrates, for the first time, multi-scale features of dengue's population biology. The implications of this research are applicable to many other pathogens, involving challenging new ways of determining the underlying causes of the complex phylodynamics of antigenically diverse pathogens.

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Orthogonal frequency division multiplexing (OFDM) is being successfully used in numerous applications. It was chosen for IEEE 802.11a wireless local area network (WLAN) standard, and it is being considered for the fourthgeneration mobile communication systems. Along with its many attractive features, OFDM has some principal drawbacks. Sensitivity to frequency errors is the most dominant of these drawbacks. In this thesis, the frequency offset and phase noise effects on OFDM based communication systems are investigated under a variety of channel conditions covering both indoor and outdoor environments. The simulation performance results of the OFDM system for these channels are presented.
Lieutenant Junior Grade, Turkish Navy

L0 penalized likelihood procedures like Mallows' Cp, AIC, and BIC directly penalize for the number of variables included in a regression model. This is a straightforward approach to the problem of overfitting, and these methods are now part of every statistician's repertoire. However, these procedures have been shown to sometimes result in unstable parameter estimates as a result on the L0 penalty's discontinuity at zero. One proposed alternative, seamless-L0 (SELO), utilizes a continuous penalty function that mimics L0 and allows for stable estimates. Like other similar methods (e.g. LASSO and SCAD), SELO produces sparse solutions because the penalty function is non-differentiable at the origin. Because these penalized likelihoods are singular (non-differentiable) at zero, there is no closed-form solution for the extremum of the objective function. We propose a continuous and everywhere-differentiable penalty function that can have arbitrarily steep slope in a neighborhood near zero, thus mimicking the L0 penalty, but allowing for a nearly closed-form solution for the beta-hat vector. Because our function is not singular at zero, beta-hat will have no zero-valued components, although some will have been shrunk arbitrarily close thereto. We employ a BIC-selected tuning parameter used in the shrinkage step to perform zero-thresholding as well. We call the resulting vector of coefficients the ShrinkSet estimator. It is comparable to SELO in terms of model performance (selecting the truly nonzero coefficients, overall MSE, etc.), but we believe it to be more intuitive and simpler to compute. We provide strong evidence that the estimator enjoys favorable asymptotic properties, including the oracle property.