We propose a model for an insurance loss index and the claims process of a single insurance company holding a fraction of the total number of contracts that captures both ordinary losses and losses due to catastrophes. In this model we price a catastrophe derivative by the method of utility indifference pricing. The associated stochastic optimization problem is treated by techniques for piecewise deterministic Markov processes. A numerical study illustrates our results.

Esta tese foca no problema de estimação de estado e de identificação de parâametros para modelos afins por partes. Modelos afins por partes são obtidos quando o domínio do estado ou da entrada do sistema e particionado em regiões e, para cada região, um submodelo linear ou afim e utilizado para descrever a dinâmica do sistema. Propomos um algoritmo para estimação recursiva de estados e um algoritmo de identificação de parâmetros para uma classe de modelos afins por partes. Propomos um estimador de estados Bayesiano que utiliza o filtro de Kalman em cada um dos submodelos. Neste estimador, a função distribuição cumulativa e utilizada para calcular a distribuição a posteriori do estado assim como a probabilidade de cada submodelo. Já o método de identificação proposto utiliza o algoritmo EM (Expectation Maximization algorithm) para identificar os parâmetros do modelo. A função distribuição cumulativa e utilizada para calcular a probabilidade de cada submodelo a partir da medida do sistema. Em seguida, utilizamos o filtro de Kalman suavizado para estimar o estado e calcular uma função substituta da função likelihood. Tal função e então utilizada para identificar os parâmetros do modelo. O estimador proposto foi utilizado para estimar o estado do modelo não linear para vibrações causadas por folgas. Foram realizadas simulações, onde comparamos o método proposto ao filtro de Kalman estendido e o filtro de partículas. O algoritmo de identificação foi utilizado para identificar os parâmetros do modelo do jato JAS 39 Gripen, assim como, o modelos não linear de vibrações causadas por folgas. / This thesis focuses on the state estimation and parameter identi cation problems of piecewise a ne models. Piecewise a ne models are obtained when the state domain or the input domain are partitioned into regions and, for each region, a linear or a ne submodel is used to describe the system dynamics. We propose a recursive state estimation algorithm and a parameter identi cation algorithm to a class of piecewise a ne models. We propose a Bayesian state estimate which uses the Kalman lter in each submodel. In the this estimator, the cumulative distribution is used to compute the posterior distribution of the state as well as the probability of each submodel. On the other hand, the proposed identi cation method uses the Expectation Maximization (EM) algorithm to identify the model parameters. We use the cumulative distribution to compute the probability of each submodel based on the system measurements. Subsequently, we use the Kalman smoother to estimate the state and compute a surrogate function for the likelihood function. This function is used to estimate the model parameters. The proposed estimator was used to estimate the state of the nonlinear model for vibrations caused by clearances. Numerical simulations were performed, where we have compared the proposed method to the extended Kalman lter and the particle lter. The identi cation algorithm was used to identify the model parameters of the JAS 39 Gripen aircraft as well as the nonlinear model for vibrations caused by clearances.

Apresentamos uma generalização para as classes das álgebras quase inclinadas e quase hereditárias, que chamamos de álgebras m-quase inclinadas e m-quase hereditárias. Para estas últimas, pode-se obter uma trissecção de suas categorias de módulos determinada pelas subcategorias L^m = {X indecomponível; dimensão projetiva de Y é menor ou igual a m, para cada antecessor Y de X} e R = {X indecomponível; dimensão injetiva de Y é menor ou igual a 1, para cada sucessor Y de X}, além de ser possível mostrar que se existe um módulo E_m de forma a obtermos a igualdade de conjuntos {X módulo; Hom(E_m, \\tau X) = 0} = {X módulo; dimensão projetiva de X é menor ou igual a m}, então E_m é soma de somandos de módulos em R e todo caminho de indecomponíveis com início em um somando E de E_m e final em um módulo projetivo pode ser refinado a um caminho de morfismos irredutíveis, que é ainda seccional. Como consequência desse resultado obtém-se que as álgebras m-quase hereditárias são caracterizadas pelo fato de que todos seus módulos projetivos pertencem a L^m. É possível verificar que toda álgebra m-quase inclinada de dimensão global m+1 é m-quase hereditária e, consequentemente, que toda álgebra hereditária por partes de tipo mod H, para alguma álgebra hereditária H, com dimensão global m+1 é m-quase hereditária. Apresentamos ainda um exemplo de uma álgebra 2-quase hereditária que não é 2-quase inclinada, não sendo válida, portanto, a recíproca do resultado acima. Buscamos, dessa forma, estabelecer condições que quando assumidas sobre uma álgebra 2-quase hereditária possam garantir que esta é 2-quase inclinada e, em particular, hereditária por partes. Recorremos, para isso, à aplicação obtida por meio de uma adaptação de resultados de Happel, Reiten e Smalo, que sob certas hipóteses permite concluir que uma álgebra é álgebra de endomorfismos de um objeto inclinante. Como resultado, mostra-se que uma álgebra 2-quase hereditária com certas outras propriedades e que satisfaz as condições (H1), (H2) e (H3) é 2-quase inclinada. / We present a generalization of the classes of quasitilted and almost hereditary algebras, which we call m-quasitilted and m-almost hereditary algebras. For the latter one, we can obtain a trisection of their module categories determined by the following subcategories L^m = {X indecomposable; projective dimension of Y is at most m for each predecessor Y of X} and R = {X indecomposable; injective dimension of Y is at most 1 for each successor Y of X}. Moreover, if there exists a module E_m such that {X; Hom(E_m, \\tau X) = 0} = {X; projective dimension of X is at most m} then E_m is a sum of direct summands of modules in R and any path of indecomposable modules starting in a module E which is a direct summand of E_m and ending in a projective module can be refined to a path of irreducible morphisms, which is also sectional. This result on paths allow us to obtain a characterization for m-almost hereditary algebras in terms of their projective modules. It is also possible to prove that any m-quasitilted algebra with global dimension m+1 is a m-almost hereditary algebra and as a consequence we can obtain that any piecewise hereditary algebra of type mod H, for some hereditary algebra H, and with global dimension m+1 is m-almost hereditary. We present an example of a 2-almost hereditary which is not 2-quasitilted, which entails that the converse of the above mentioned result does not hold true. Thus we seek for conditions which can ensure that a given 2-almost hereditary is 2-quasitilted and, in particular, a piecewise hereditary algebra. For this, we use the correspondence obtained as an adaptation of results of Happel, Reiten and Smalo, which under certain assumptions shows that an algebra is an endomorphism algebra of a tilting object. It is shown that a 2-almost hereditary algebra with some other properties and satisfying (H1), (H2) and (H3) is 2-quasitilted.

After introducing concepts from convex analysis, we study how to continuously transform one convex function into another. A natural choice is the arithmetic average, as it is pointwise continuous; however, this choice fails to average functions with different domains. On the contrary, the proximal average is not only continuous (in the epi-topology) but can actually average functions with disjoint domains. In fact, the proximal average not only inherits strict convexity (like the arithmetic average) but also inherits smoothness and differentiability (unlike the arithmetic average). Then we introduce a computational framework for computer-aided convex analysis. Motivated by the proximal average, we notice that the class of piecewise linear-quadratic (PLQ) functions is closed under (positive) scalar multiplication, addition, Fenchel conjugation, and Moreau envelope. As a result, the PLQ framework gives rise to linear-time and linear-space algorithms for convex PLQ functions. We extend this framework to nonconvex PLQ functions and present an explicit convex hull algorithm. Finally, we discuss a method to find primal-dual symmetric antiderivatives from cyclically monotone operators. As these antiderivatives depend on the minimal and maximal Rockafellar functions [5, Theorem 3.5, Corollary 3.10], it turns out that the minimal and maximal function in [12, p.132,p.136] are indeed the same functions. Algorithms used to compute these antiderivatives can be formulated as shortest path problems.

Interval censoring arises frequently in life history data, as individuals are often only observed at a sequence of assessment times. This leads to a situation where we do not know when an event of interest occurs, only that it occurred somewhere between two assessment times. Here, the focus will be on methods of estimation for recurrent event data, current status data, and multistate data, subject to interval censoring. With recurrent event data, the focus is often on estimating the rate and mean functions. Nonparametric estimates are readily available, but are not smooth. Methods based on local likelihood and the assumption of a Poisson process are developed to obtain smooth estimates of the rate and mean functions without specifying a parametric form. Covariates and extra-Poisson variation are accommodated by using a pseudo-profile local likelihood. The methods are assessed by simulations and applied to a number of datasets, including data from a psoriatic arthritis clinic. Current status data is an extreme form of interval censoring that occurs when each individual is observed at only one assessment time. If current status data arise in clusters, this must be taken into account in order to obtain valid conclusions. Copulas offer a convenient framework for modelling the association separately from the margins. Estimating equations are developed for estimating marginal parameters as well as association parameters. Efficiency and robustness to the choice of copula are examined for first and second order estimating equations. The methods are applied to data from an orthopedic surgery study as well as data on joint damage in psoriatic arthritis. Multistate models can be used to characterize the progression of a disease as individuals move through different states. Considerable attention is given to a three-state model to characterize the development of a back condition known as spondylitis in psoriatic arthritis, along with the associated risk of mortality. Robust estimates of the state occupancy probabilities are derived based on a difference in distribution functions of the entry times. A five-state model which differentiates between left-side and right-side spondylitis is also considered, which allows us to characterize what effect spondylitis on one side of the body has on the development of spondylitis on the other side. Covariate effects are considered through multiplicative time homogeneous Markov models. The robust state occupancy probabilities are also applied to data on CMV infection in patients with HIV.

Interval censoring arises frequently in life history data, as individuals are often only observed at a sequence of assessment times. This leads to a situation where we do not know when an event of interest occurs, only that it occurred somewhere between two assessment times. Here, the focus will be on methods of estimation for recurrent event data, current status data, and multistate data, subject to interval censoring. With recurrent event data, the focus is often on estimating the rate and mean functions. Nonparametric estimates are readily available, but are not smooth. Methods based on local likelihood and the assumption of a Poisson process are developed to obtain smooth estimates of the rate and mean functions without specifying a parametric form. Covariates and extra-Poisson variation are accommodated by using a pseudo-profile local likelihood. The methods are assessed by simulations and applied to a number of datasets, including data from a psoriatic arthritis clinic. Current status data is an extreme form of interval censoring that occurs when each individual is observed at only one assessment time. If current status data arise in clusters, this must be taken into account in order to obtain valid conclusions. Copulas offer a convenient framework for modelling the association separately from the margins. Estimating equations are developed for estimating marginal parameters as well as association parameters. Efficiency and robustness to the choice of copula are examined for first and second order estimating equations. The methods are applied to data from an orthopedic surgery study as well as data on joint damage in psoriatic arthritis. Multistate models can be used to characterize the progression of a disease as individuals move through different states. Considerable attention is given to a three-state model to characterize the development of a back condition known as spondylitis in psoriatic arthritis, along with the associated risk of mortality. Robust estimates of the state occupancy probabilities are derived based on a difference in distribution functions of the entry times. A five-state model which differentiates between left-side and right-side spondylitis is also considered, which allows us to characterize what effect spondylitis on one side of the body has on the development of spondylitis on the other side. Covariate effects are considered through multiplicative time homogeneous Markov models. The robust state occupancy probabilities are also applied to data on CMV infection in patients with HIV.

Inference of regulatory relations in dynamical systems is a promising active research area. Recently, most of the investigations in this field have been stimulated by the researches in functional genomics. In this thesis, the inferential modeling problem for switching hybrid systems is studied. The hybrid systems refers to dynamical systems in which discrete and continuous variables regulate each other, in other words the jumps and flows are interrelated. In this study, piecewise linear approximations are used for modeling purposes and it is shown that piecewise linear models are capable of displaying the evolutionary characteristics of switching hybrid systems approxi- mately. For the mentioned systems, detection of switching instances and inference of locally linear parameters from empirical data provides a solid understanding about the system dynamics. Thus, the inference methodology is based on these issues. The primary difference of the inference algorithm is the idea of transforming the switch- ing detection problem into a jump detection problem by derivative estimation from discrete data. The jump detection problem has been studied extensively in signal processing literature. So, related techniques in the literature has been analyzed care- fully and suitable ones adopted in this thesis. The primary advantage of proposed method would be its robustness in switching detection and derivative estimation. The theoretical background of this robustness claim and the importance of robustness for real world applications are explained in detail.

Many dynamical systems in nature and technology involve delays in the interaction of variables forming the system. Furthermore, many of such systems involve external inputs or perturbations which might force the system to have arbitrary initial function. The conventional way to model these systems is using delay differential equations (DDE). However, DDEs with arbitrary initial functions has serious problems for finding analytical and computational solutions. This fact is a strong motivation for considering abstractions and approximations for dynamical systems involving delay. In this thesis, the piecewise linear systems with delay on piecewise constant part which is a useful subclass of hybrid dynamical systems is studied. We introduced various representations of these systems and studied the state transition conditions. We showed that there exists fixed point and periodic stable solutions. We modelled the genomic regulation of fission yeast cell cycle. We discussed various potential uses including approximating the DDEs and finally we concluded.

Regulatory processes and history dependent behavior appear in many dynamical systems in nature and technology. For modeling regulatory processes, hybrid systems offer several advances. From this point of view, to observe the capability of hybrid systems in a history dependent system is a strong motivation. In this thesis, we developed functional hybrid systems which exhibit memory dependent behavior such that the dynamics of the system is determined by both the location of the state vector and the memory. This property was explained by various examples. We used the hybrid system with memory in modeling the gene regulatory network of human immune response to Influenza A virus infection. We investigated the sensitivity of the piecewise linear model with memory. We introduced how the model can be developed in future.

After introducing concepts from convex analysis, we study how to continuously transform one convex function into another. A natural choice is the arithmetic average, as it is pointwise continuous; however, this choice fails to average functions with different domains. On the contrary, the proximal average is not only continuous (in the epi-topology) but can actually average functions with disjoint domains. In fact, the proximal average not only inherits strict convexity (like the arithmetic average) but also inherits smoothness and differentiability (unlike the arithmetic average). Then we introduce a computational framework for computer-aided convex analysis. Motivated by the proximal average, we notice that the class of piecewise linear-quadratic (PLQ) functions is closed under (positive) scalar multiplication, addition, Fenchel conjugation, and Moreau envelope. As a result, the PLQ framework gives rise to linear-time and linear-space algorithms for convex PLQ functions. We extend this framework to nonconvex PLQ functions and present an explicit convex hull algorithm. Finally, we discuss a method to find primal-dual symmetric antiderivatives from cyclically monotone operators. As these antiderivatives depend on the minimal and maximal Rockafellar functions [5, Theorem 3.5, Corollary 3.10], it turns out that the minimal and maximal function in [12, p.132,p.136] are indeed the same functions. Algorithms used to compute these antiderivatives can be formulated as shortest path problems.