We formalize and analyze the notions of monotonicity and complete monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide in continuoustime but not in discrete-time.

This thesis consists of three papers devoted to the study of monotonicity formulas and their applications in elliptic and parabolic free boundary problems. The first paper concerns an inhomogeneous parabolic problem. We obtain global and local almost monotonicity formulas and apply one of them to show a regularity result of a problem that arises in connection with continuation of heat potentials.In the second paper, we consider an elliptic two-phase problem with coefficients bellow the Lipschitz threshold. Optimal $C^{1,1}$ regularity of the solution and a regularity result of the free boundary are established.The third and last paper deals with a parabolic free boundary problem with Hölder continuous coefficients. Optimal $C^{1,1}\cap C^{0,1}$ regularity of the solution is proven. / QC20100621

The stochastic Galerkin and collocation methods are used to solve an advection-diusion equation with uncertain and spatially varying viscosity. We investigate well-posedness, monotonicity and stability for the extended system resulting from the Galerkin projection of the advection-diusion equation onto the stochastic basis functions. High-order summationby- parts operators and weak imposition of boundary conditions are used to prove stability of the semi-discrete system. It is essential that the eigenvalues of the resulting viscosity matrix of the stochastic Galerkin system are positive and we investigate conditions for this to hold. When the viscosity matrix is diagonalizable, stochastic Galerkin and stochastic collocation are similar in terms of computational cost, and for some cases the accuracy is higher for stochastic Galerkin provided that monotonicity requirements are met. We also investigate the total spatial operator of the semi-discretized system and its impact on the convergence to steadystate

Lookback option is a well-known path-dependent option where its payoff depends on the historical extremum prices. The thesis focuses on the binomial pricing of the American floating strike lookback put options with payoff at time $t$ (if exercise) characterized by \[ \max_{k=0, \ldots, t} S_k - S_t, \] where $S_t$ denotes the price of the underlying stock at time $t$. Build upon the idea of \hyperlink{RBCV}{Reiner Babbs Cheuk and Vorst} (RBCV, 1992) who proposed a transformed binomial lattice model for efficient pricing of this class of option, this thesis extends and enhances their binomial recursive algorithm by exploiting the additional combinatorial properties of the lattice structure. The proposed algorithm is not only computational efficient but it also significantly reduces the memory constraint. As a result, the proposed algorithm is more than 1000 times faster than the original RBCV algorithm and it can compute a binomial lattice with one million time steps in less than two seconds. This algorithm enables us to extrapolate the limiting (American) option value up to 4 or 5 decimal accuracy in real time.

In this thesis, we extend the spectral collocation methods(SCM) (i.e., pseudo-spectral method) in Quarteroni and Valli [27] for the semilinear, parameter-dependentproblems(PDP) in the square with the Dirichlet boundary condition. The optimal error bounds are derived in this thesis for both H1 and L2 norms. For the solutions sufficiently smooth, the very high convergence rates can be obtained. The algorithms of the SCM are simple and easy to carry out. Only a few of basis functions are needed so that not only can the high accuracy of the PDP solutions be achieved, but also a great deal of CPU time may be saved. Moreover, for PDP the stability analysis of SCM is also made, to have the same growth rates of condition number as those for Poisson¡¦s equation. Numerical experiments are carried out to verify the theoretical analysis made.

Lookback option is a well-known path-dependent option where its payoff depends on the historical extremum prices. The thesis focuses on the binomial pricing of the American floating strike lookback put options with payoff at time $t$ (if exercise) characterized by \[ \max_{k=0, \ldots, t} S_k - S_t, \] where $S_t$ denotes the price of the underlying stock at time $t$. Build upon the idea of \hyperlink{RBCV}{Reiner Babbs Cheuk and Vorst} (RBCV, 1992) who proposed a transformed binomial lattice model for efficient pricing of this class of option, this thesis extends and enhances their binomial recursive algorithm by exploiting the additional combinatorial properties of the lattice structure. The proposed algorithm is not only computational efficient but it also significantly reduces the memory constraint. As a result, the proposed algorithm is more than 1000 times faster than the original RBCV algorithm and it can compute a binomial lattice with one million time steps in less than two seconds. This algorithm enables us to extrapolate the limiting (American) option value up to 4 or 5 decimal accuracy in real time.

Στη παρούσα εργασία ασχοληθήκαμε με ιδιότητες μονοτονίας των Τροποποιημένων συναρτήσεων Bessel 1ου και 2ου είδους. Συγκεκριμένα ομαδοποιήσαμε ήδη υπάρχοντα φράγματα για τα κλάσματα των συναρτήσεων αυτών. Η εύρεση φραγμάτων για τα κλάσματα των Τροποποιημένων Συναρτήσεων Bessel είναι σημαντική, λόγω της χρησιμότητάς τους σε διάφορους κλάδους των Μαθηματικών και όχι μόνο, όπως ενδεικτικά, στην Πεπερασμένη Ελαστικότητα, στην Στατιστική και στις Πιθανότητες, στην Ειδική Θεωρία Σχετικότητας, στην Μηχανική των Ρευστών, στην Ηλεκτρομηχανική, στη Βιοφυσική, στη Μαθηματική Φυσική και αλλού. Αρχικά, στο Κεφάλαιο 1, παρατέθηκαν κάποια βασικά στοιχεία, όπως ορισμοί των συναρτήσεων Bessel 1ου και 2ου είδους (Τροποποιημένων και μη) και αναδρομικές σχέσεις που ικανοποιούν. Στο Κεφάλαιο 2, γίνεται η καταγραφή και σύγκριση άνω και κάτω φραγμάτων για τα διάφορα κλάσματα των Τροποποιημένων συναρτήσεων Bessel 1ου είδους, καθώς και αναφορά σε ανισότητες τύπου Turán για τις συναρτήσεις αυτές. Επίσης, αναφέρεται η μεθοδολογία στην οποία στηρίχθηκε ο κάθε ερευνητής για να πάρει τα αντίστοιχα αποτελέσματα. Στο Κεφάλαιο 3, γίνεται η αντίστοιχη διαδικασία για τα κλάσματα και εκ νέου αναφορά σε ανισότητες τύπου Turán για αυτές τις συναρτήσεις. / In this project we described properties of Modified Bessel functions of the 1st and 2nd kind. Specifically we have grouped existing bounds for the quotients of these functions. These bounds of the Modified Bessel functions is very importand and could be found in different branches of Mathematics and other sciences, such as in Finite Elasticity, in Statistics and Probability Theory, in Relativity Theory, in Fluid Mechanics, in Engineering, in Biophysics, in Mathematical Physics and so on. Firsty, in Chapter 1, we cited some basic data, such as definitions of definitions of Bessel fynctions of the 1st and 2nd kind (both simple and Modified) and recurrence relations that they satisfy. In Chapter 2, we describe upper and lower bounds of different quotients of Modified Bessel functions of the 1st kind and reference to Turán type Inequalities of those functions. Moreover, we refer to the method that each recearcher based on in order to prove the required results. In Chapter 3, we have the same process but for Modified Bessel functons of the 2nd kind as well as reference to Turán type Inequalities for the corresponding functions.

This thesis is concerned with the formulation of a suitable notion of monotonicity of discrete and continuous-time dynamical systems on Lie groups and homogeneous spaces. In a linear space, monotonicity refers to the property of a system that preserves an ordering of the elements of the space. Monotone systems have been studied in detail and are of great interest for their numerous applications, as well as their close connections to many physical and biological systems. In a linear space, a powerful local characterisation of monotonicity is provided by differential positivity with respect to a constant cone field, which combines positivity theory with a local analysis of nonlinear systems. Since many dynamical systems are naturally defined on nonlinear spaces, it is important to seek a suitable adaptation of monotonicity on such spaces. However, the question of how one can develop a suitable notion of monotonicity on a nonlinear manifold is complicated by the general absence of a clear and well-defined notion of order on such a space. Fortunately, for Lie groups and important examples of homogeneous spaces that are ubiquitous in many problems of engineering and applied mathematics, symmetry provides a way forward. Specifically, the existence of a notion of geometric invariance on such spaces allows for the generation of invariant cone fields, which in turn induce notions of conal orders. We propose differential positivity with respect to invariant cone fields as a natural and powerful generalisation of monotonicity to nonlinear spaces and develop the theory in this thesis. We illustrate the ideas with numerous examples and apply the theory to a number of areas, including the theory of consensus on Lie groups and order theory on the set of positive definite matrices.

O conceito de taxa de variação é muito útil no estudo das funções reais de uma variável real e, por ser elementar, pode ser apresentado e discutido no Ensino Médio. A proposta deste trabalho é fazer um estudo detalhado e criterioso do uso da taxa de variação no estudo das funções afins, quadráticas e cúbicas. Com isso, pretende-se desenvolver resultados e apresentar diversas aplicações deste conceito a estas classes de funções que incluem, por exemplo, novas formas de obtenção dos seus gráficos, a interpretação geométrica de seus coeficientes, bem como resolver problemas simples e interessantes de otimização. Quanto às funções quadráticas, como aplicação do estudo feito sobre estas funções, apresentamos uma maneira matematicamente rigorosa de conceituar o Movimento Retilíneo Uniformemente Variado (MRUV), bem como, obtemos as equações horárias deste movimento. A definição que apresentamos é equivalente a usual, mas não faz uso dos Cálculos Infinitesimal e Integral. Como nosso objetivo é eminentemente matemático, não apresentamos qualquer interpretação Física deste conceito na forma que o introduzimos. Já o estudo das funções cúbicas pode ser visto como uma interessante aplicação das funções quadráticas, uma vez que, através do uso das taxas de variação, a descrição (traçado do gráfico) de uma função cúbica é reduzida ao estudo do sinal de uma função quadrática associada. / The concept of rate of change is very useful in the study of real functions of one real variable. Since this is a simple concept it can be presented and discussed in high school. The purpose of this work is to make a comprehensive study use of the rate of change in the study of the affine, quadratic and cubic functions. We prove results and present several applications of this concept to these classes of functions that include, for example, new ways of obtaining their graphs, geometric interpretation of their coefficients, and the solution of simple and interesting optimization problems. As for the quadratic functions, as an application of our study about these functions, we obtain in a mathematically rigorous definition of Uniformly Varied Rectilinear Motion and obtain the equations of the motion without using the Infinitesimal or Integral Calculus. Since we were essentially concerned with the mathematical aspect of this phenomenon, we did not present a discussion about the physical interpretation of our definition. The study of cubic functions can be seen as an interesting application of the quadratic functions, since through the use of rate changing, the description of the graph of a cubic function is reduced to the study of the sign of a quadratic function associated to the given cubic.

O conceito de taxa de variação é muito útil no estudo das funções reais de uma variável real e, por ser elementar, pode ser apresentado e discutido no Ensino Médio. A proposta deste trabalho é fazer um estudo detalhado e criterioso do uso da taxa de variação no estudo das funções afins, quadráticas e cúbicas. Com isso, pretende-se desenvolver resultados e apresentar diversas aplicações deste conceito a estas classes de funções que incluem, por exemplo, novas formas de obtenção dos seus gráficos, a interpretação geométrica de seus coeficientes, bem como resolver problemas simples e interessantes de otimização. Quanto às funções quadráticas, como aplicação do estudo feito sobre estas funções, apresentamos uma maneira matematicamente rigorosa de conceituar o Movimento Retilíneo Uniformemente Variado (MRUV), bem como, obtemos as equações horárias deste movimento. A definição que apresentamos é equivalente a usual, mas não faz uso dos Cálculos Infinitesimal e Integral. Como nosso objetivo é eminentemente matemático, não apresentamos qualquer interpretação Física deste conceito na forma que o introduzimos. Já o estudo das funções cúbicas pode ser visto como uma interessante aplicação das funções quadráticas, uma vez que, através do uso das taxas de variação, a descrição (traçado do gráfico) de uma função cúbica é reduzida ao estudo do sinal de uma função quadrática associada. / The concept of rate of change is very useful in the study of real functions of one real variable. Since this is a simple concept it can be presented and discussed in high school. The purpose of this work is to make a comprehensive study use of the rate of change in the study of the affine, quadratic and cubic functions. We prove results and present several applications of this concept to these classes of functions that include, for example, new ways of obtaining their graphs, geometric interpretation of their coefficients, and the solution of simple and interesting optimization problems. As for the quadratic functions, as an application of our study about these functions, we obtain in a mathematically rigorous definition of Uniformly Varied Rectilinear Motion and obtain the equations of the motion without using the Infinitesimal or Integral Calculus. Since we were essentially concerned with the mathematical aspect of this phenomenon, we did not present a discussion about the physical interpretation of our definition. The study of cubic functions can be seen as an interesting application of the quadratic functions, since through the use of rate changing, the description of the graph of a cubic function is reduced to the study of the sign of a quadratic function associated to the given cubic.