<p>This thesis deals with a model of nonlinearly perturbed continuous-time renewal equation with nonpolynomial perturbations. The characteristics, namely the defect and moments, of the distribution function generating the renewal equation are assumed to have expansions with respect to a non-polynomial asymptotic scale: $\{\varphi_{\nn} (\varepsilon) =\varepsilon^{\nn \cdot \w}, \nn \in \mathbf{N}_0^k\}$  as $\varepsilon \to 0$, where $\mathbf{N}_0$ is the set of non-negative integers, $\mathbf{N}_0^k \equiv \mathbf{N}_0 \times \cdots \times \mathbf{N}_0, 1\leq k <\infty$ with the product being taken $k$ times and $\w$ is a $k$ dimensional parameter vector that satisfies certain properties. For the one-dimensional case, i.e., $k=1$, this model reduces to the model of nonlinearly perturbed renewal equation with polynomial perturbations which is well studied in the literature.  The goal of the present study is to obtain the exponential asymptotics for the solution to the perturbed renewal equation in the form of exponential asymptotic expansions and present possible applications.</p><p>The thesis is based on three papers which study successively the model stated above. Paper A investigates the two-dimensional case, i.e. where $k=2$. The corresponding asymptotic exponential expansion for the solution to the perturbed renewal equation is given. The asymptotic results are applied to an example of the perturbed risk process, which leads to diffusion approximation type asymptotics for the ruin probability.  Numerical experimental studies on this example of perturbed risk process are conducted in paper B, where Monte Carlo simulation are used to study the accuracy and properties of the asymptotic formulas. Paper C presents the asymptotic results for the more general case where the dimension $k$ satisfies $1\leq k <\infty$, which are applied to the asymptotic analysis of the ruin probability in an example of perturbed risk processes with this general type of non-polynomial perturbations.  All the proofs of the theorems stated in paper C are collected in its supplement: paper D.</p>

This thesis deals with a model of nonlinearly perturbed continuous-time renewal equation with nonpolynomial perturbations. The characteristics, namely the defect and moments, of the distribution function generating the renewal equation are assumed to have expansions with respect to a non-polynomial asymptotic scale: $\{\varphi_{\nn} (\varepsilon) =\varepsilon^{\nn \cdot \w}, \nn \in \mathbf{N}_0^k\}$  as $\varepsilon \to 0$, where $\mathbf{N}_0$ is the set of non-negative integers, $\mathbf{N}_0^k \equiv \mathbf{N}_0 \times \cdots \times \mathbf{N}_0, 1\leq k &lt;\infty$ with the product being taken $k$ times and $\w$ is a $k$ dimensional parameter vector that satisfies certain properties. For the one-dimensional case, i.e., $k=1$, this model reduces to the model of nonlinearly perturbed renewal equation with polynomial perturbations which is well studied in the literature.  The goal of the present study is to obtain the exponential asymptotics for the solution to the perturbed renewal equation in the form of exponential asymptotic expansions and present possible applications. The thesis is based on three papers which study successively the model stated above. Paper A investigates the two-dimensional case, i.e. where $k=2$. The corresponding asymptotic exponential expansion for the solution to the perturbed renewal equation is given. The asymptotic results are applied to an example of the perturbed risk process, which leads to diffusion approximation type asymptotics for the ruin probability.  Numerical experimental studies on this example of perturbed risk process are conducted in paper B, where Monte Carlo simulation are used to study the accuracy and properties of the asymptotic formulas. Paper C presents the asymptotic results for the more general case where the dimension $k$ satisfies $1\leq k &lt;\infty$, which are applied to the asymptotic analysis of the ruin probability in an example of perturbed risk processes with this general type of non-polynomial perturbations.  All the proofs of the theorems stated in paper C are collected in its supplement: paper D.

Neste trabalho investigamos, dos pontos de vistas analítico e numérico, o comportamento assintótico da solução da equação do aerofólio, com uma singularidade do tipo Cauchy, de nida sobre um intervalo com uma pequena abertura. Exibimos um modelo matemático com uma solução f" para o intervalo disjunto G" = (−1,−ε) ∪ (ε, 1) e uma solução f0 que corresponde ao limite de f" quando (ε → 0), relacionando esta última com a solução da equação do aerofólio f no intervalo (−1, 1). Além do mais, demonstramos casos particulares de funções ψ = Tm e ψ = Un(onde Tm e Un são os polinômios de Tchebychev do primeiro e segundo tipo respectivamente) em que temos a igualdade f = f0 e conseqüentemente f" ≈ f. Apresentamos e comparamos numericamente as soluções f", f0 e f para diferentes funções ψ e valores de ε no intervalo G". Mostramos ainda soluções quase polinomiais analíticas da equação do aerofólio, e propomos um método espectral para a equação do aerofólio generalizada. Por m, obtemos soluções analíticas das equações do aerofólio para os intervalos G", (−1, 1)\ {0} e (−1, 1) para diferentes funções ψ(t) através da expansão em série da densidade da integral singular com núcleo Cauchy. / In this work we investigate, of the analytical and numerical points of views, the asymptotic behavior of the airfoil equation solution with a singularity of the Cauchy type, de ned over a interval with a small opening. We display a mathematical model with a f" solution to the disjoint interval G" = (−1,−ε)∪(ε, 1) and a f0 solution corresponding to limit of f" when (ε → 0), linking the latter with the solution of the airfoil equation f in the interval (−1, 1). Furthermore, we demonstrate particular cases of functions ψ = Tm and ψ = Un (where Tm and Un are the Chebyshev polynomials of the rst and second type respectively) where we have equality f = f0 and then f" ≈ f. We present and compare numerically the solutions f", f0 and f for di erent functions ψ and values of ε in G". We also show almost polynomial analytical solutions for the airfoil equation, and we propose a spectral method for the generalized airfoil equation. Finally, we obtain analytical solutions of the airfoil equations to the interval G", (−1, 1)\ {0} and (−1, 1) for various functions ψ(t) by expanding in series the density of the Cauchy singular integral.

Neste trabalho investigamos, dos pontos de vistas analítico e numérico, o comportamento assintótico da solução da equação do aerofólio, com uma singularidade do tipo Cauchy, de nida sobre um intervalo com uma pequena abertura. Exibimos um modelo matemático com uma solução f" para o intervalo disjunto G" = (−1,−ε) ∪ (ε, 1) e uma solução f0 que corresponde ao limite de f" quando (ε → 0), relacionando esta última com a solução da equação do aerofólio f no intervalo (−1, 1). Além do mais, demonstramos casos particulares de funções ψ = Tm e ψ = Un(onde Tm e Un são os polinômios de Tchebychev do primeiro e segundo tipo respectivamente) em que temos a igualdade f = f0 e conseqüentemente f" ≈ f. Apresentamos e comparamos numericamente as soluções f", f0 e f para diferentes funções ψ e valores de ε no intervalo G". Mostramos ainda soluções quase polinomiais analíticas da equação do aerofólio, e propomos um método espectral para a equação do aerofólio generalizada. Por m, obtemos soluções analíticas das equações do aerofólio para os intervalos G", (−1, 1)\ {0} e (−1, 1) para diferentes funções ψ(t) através da expansão em série da densidade da integral singular com núcleo Cauchy. / In this work we investigate, of the analytical and numerical points of views, the asymptotic behavior of the airfoil equation solution with a singularity of the Cauchy type, de ned over a interval with a small opening. We display a mathematical model with a f" solution to the disjoint interval G" = (−1,−ε)∪(ε, 1) and a f0 solution corresponding to limit of f" when (ε → 0), linking the latter with the solution of the airfoil equation f in the interval (−1, 1). Furthermore, we demonstrate particular cases of functions ψ = Tm and ψ = Un (where Tm and Un are the Chebyshev polynomials of the rst and second type respectively) where we have equality f = f0 and then f" ≈ f. We present and compare numerically the solutions f", f0 and f for di erent functions ψ and values of ε in G". We also show almost polynomial analytical solutions for the airfoil equation, and we propose a spectral method for the generalized airfoil equation. Finally, we obtain analytical solutions of the airfoil equations to the interval G", (−1, 1)\ {0} and (−1, 1) for various functions ψ(t) by expanding in series the density of the Cauchy singular integral.

Neste trabalho investigamos, dos pontos de vistas analítico e numérico, o comportamento assintótico da solução da equação do aerofólio, com uma singularidade do tipo Cauchy, de nida sobre um intervalo com uma pequena abertura. Exibimos um modelo matemático com uma solução f" para o intervalo disjunto G" = (−1,−ε) ∪ (ε, 1) e uma solução f0 que corresponde ao limite de f" quando (ε → 0), relacionando esta última com a solução da equação do aerofólio f no intervalo (−1, 1). Além do mais, demonstramos casos particulares de funções ψ = Tm e ψ = Un(onde Tm e Un são os polinômios de Tchebychev do primeiro e segundo tipo respectivamente) em que temos a igualdade f = f0 e conseqüentemente f" ≈ f. Apresentamos e comparamos numericamente as soluções f", f0 e f para diferentes funções ψ e valores de ε no intervalo G". Mostramos ainda soluções quase polinomiais analíticas da equação do aerofólio, e propomos um método espectral para a equação do aerofólio generalizada. Por m, obtemos soluções analíticas das equações do aerofólio para os intervalos G", (−1, 1)\ {0} e (−1, 1) para diferentes funções ψ(t) através da expansão em série da densidade da integral singular com núcleo Cauchy. / In this work we investigate, of the analytical and numerical points of views, the asymptotic behavior of the airfoil equation solution with a singularity of the Cauchy type, de ned over a interval with a small opening. We display a mathematical model with a f" solution to the disjoint interval G" = (−1,−ε)∪(ε, 1) and a f0 solution corresponding to limit of f" when (ε → 0), linking the latter with the solution of the airfoil equation f in the interval (−1, 1). Furthermore, we demonstrate particular cases of functions ψ = Tm and ψ = Un (where Tm and Un are the Chebyshev polynomials of the rst and second type respectively) where we have equality f = f0 and then f" ≈ f. We present and compare numerically the solutions f", f0 and f for di erent functions ψ and values of ε in G". We also show almost polynomial analytical solutions for the airfoil equation, and we propose a spectral method for the generalized airfoil equation. Finally, we obtain analytical solutions of the airfoil equations to the interval G", (−1, 1)\ {0} and (−1, 1) for various functions ψ(t) by expanding in series the density of the Cauchy singular integral.

<div> <br></div><div> We consider the behavior of zeros of random polynomials of the from</div><div> \begin{equation*}</div><div> P_{n,m}(z) := \eta_0\varphi_m^{(m)}(z) + \eta_1 \varphi_{m+1}^{(m)}(z) + \cdots + \eta_n \varphi_{n+m}^{(m)}(z)</div><div> \end{equation*}</div><div> as \( n\to\infty \), where \( m \) is a non-negative integer (most of the work deal with the case \( m =0 \) ), \( \{\eta_n\}_{n=0}^\infty \) is a sequence of i.i.d. Gaussian random variables, and \( \{\varphi_n(z)\}_{n=0}^\infty \) is a sequence of orthonormal polynomials on the unit circle \( \mathbb T \) for some Borel measure \( \mu \) on \( \mathbb T \) with infinitely many points in its support. Most of the work is done by manipulating the density function for the expected number of zeros of a random polynomial, which we call the intensity function.</div>

In this thesis, we consider the pricing problem of an American put option. We introduce a new market model for the evolution of the underlying asset price. Our model adds a new parameter to the well known Heston model. Hence we name our model the extended Heston model. To solve the American put pricing problem we adapt the idea developed by Fouque et al. (2000) to derive the asymptotic formula. We then connect the idea developed by Medvedev and Scaillet (2010) to provide an asymptotic solution for the leading order term P0. We do numerical analysis to gain insight into the accuracy and validity of our asymptotic approximation formula.

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Cette thèse traite de la diffusion par des surfaces rugueuses monodimensionnelles. Les surfaces présentant des petites échelles de variations nécessitent une discrétisation fine pour représenter les effets de diffusion sur le champ diffracté, ce qui augmente les coûts numériques. Deux aspects sont considérés : la réduction de la taille du problème en construisant une condition aux limiteséquivalente traduisant les effets des variations rapides et la réduction du nombre d’itérations nécessaires pour résoudre le système linéaire issu de la méthode des moments par une méthode basée sur les sous-espaces de Krylov. En ce qui concerne la réduction de la taille du problème, une technique d’homogénéisation est utilisée pour transformer la condition aux limites posée sur lasurface rugueuse par des paramètres effectifs. Ces paramètres sont déterminés par des problèmes auxiliaires qui tiennent compte des échelles fines de la surface. Dans le cas de surfaces parfaitement métalliques, la procédure est appliquée en polarisation Transverse Magnétique (TM) et Transverse Électrique (TE). Une impédance équivalente de Léontovich d’ordre 1 est déduite.Le procédure est automatique et les ordres supérieurs sont dérivés pour la polarisation TM. La procédure d’homogénéisation est aussi appliquée pour des interfaces rugueuses séparant deux milieux diélectriques. En ce qui concerne la réduction du nombre d’itérations, un préconditionneur, basé sur des considérations physiques, est construit à partir des modes de Floquet. Bien que le préconditionneur soit initialement élaboré pour des surfaces périodiques, nous montrons qu’il est aussi efficace pour des surfaces tronquées éclairées par une onde plane. L’efficacité des deux aspects présentés dans cette thèse est numériquement illustrée pour des configurations d’intérêt. / This work is about the scattering by monodimensional rough surfaces. Surfaces presenting small scales of variations need a very refined mesh to finally capture the scattering field behaviour what increases the computational cost. Two aspects are considered : the reduction of the problemsize through an effective boundary condition incorporating the effect of rapid variations and the reduction of the number of iterations to solve the linear system arising from method of moments by a method based on Krylov subspace. Firstly, an homogenization process is used to convert the boundary condition on the rough interface into effective parameters. These parameters are determined by the solutions of auxiliary problems which involve the detailed profile of the interface. In the case of perfectly metallic surfaces, the process is applied to the E- and H-polarization and an Leontovich impedance of order 1 is deduced. The process is automatic and higher orders are derived for E-polarization. The homogenization process is also applied to dielectric rough interfaces. Secondly, a physically-based preconditioner is built with Floquet’s modes. Although the preconditioner has been designed for periodical surfaces, it was shown to be efficient in the case of truncated surfaces illuminated by a plane wave. The efficiency of both aspects is numerically illustrated for some configurations of interest.