The zero-noise limit of differential equations with singular coefficients is investigated for the first time in the case when the noise is a general alpha-stable process. It is proved that extremal solutions are selected and the probability of selection is computed. Detailed analysis of the characteristic function of an exit time form on the half-line is performed, with a suitable decomposition in small and large jumps adapted to the singular drift.

Stochastic di®erential equations (SDEs) are　central to much of modern finance theory and have been widely used to model the behaviour of key variables such as the　instantaneous short-term interest rate, asset prices, asset returns and their volatility.　The explanatory and/or predictive power of these models depends crucially on the particularisation of the model SDE(s) to real data through the choice of values for their parameters. In econometrics, optimal parameter estimates are generally considered to　be those that maximise the likelihood of the sample. In the context of the estimation of the parameters of SDEs, however, a closed-form expression for the likelihood　function is rarely available and hence exact maximum-likelihood (EML) estimation is usually infeasible. The key research problem examined in this thesis is the development of generic, accurate and computationally feasible estimation procedures based on the　ML principle, that can be implemented in the absence of a closed-form expression for　the likelihood function. The overall recommendation to come out of the thesis is that　an estimation procedure based on the finite-element solution of a reformulation of the Fokker-Planck equation in terms of the transitional cumulative distribution function(CDF) provides the best balance across all of the desired characteristics. The recommended approach involves the use of an interpolation technique proposed in this thesis which greatly reduces the required computational effort.

In this thesis we use techniques from white noise analysis to study solutions of semilinear stochastic differential equations in a Hilbert space H: {dX[subscript]t = (AX[subscript]t + F(t,X[subscript]t)) dt + ơ(t,X[subscript]t) δB[subscript]t, t∈ (0,T], X[subscript]0 = ξ, where A is a generator of either a C[subscript]0-semigroup or an n-times integrated semigroup, and B is a cylindrical Wiener process. We then consider applications to forward interest rate models, such as in the Heath-Jarrow-Morton framework. We also reformulate a phenomenological model of the forward rate. / Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Science, 2009

In this thesis we use techniques from white noise analysis to study solutions of semilinear stochastic differential equations in a Hilbert space H: {dX[subscript]t = (AX[subscript]t + F(t,X[subscript]t)) dt + ơ(t,X[subscript]t) δB[subscript]t, t∈ (0,T], X[subscript]0 = ξ, where A is a generator of either a C[subscript]0-semigroup or an n-times integrated semigroup, and B is a cylindrical Wiener process. We then consider applications to forward interest rate models, such as in the Heath-Jarrow-Morton framework. We also reformulate a phenomenological model of the forward rate. / Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Science, 2009

Diss. Uppsala : Univ., 2004.

This thesis consists of several problems concerning numerical approximations for stochastic differential equations, and is divided into three parts. The first one is on the integrability and asymptotic stability with respect to a certain class of Lyapunov functions, and the preservation of the comparison theorem for the explicit numerical schemes. In general, those properties of the original equation can be lost after discretisation, but it will be shown that by some suitable modification of the Euler scheme they can be preserved to some extent while keeping the strong convergence rate maintained. The second part focuses on the approximation of iterated stochastic integrals, which is the essential ingredient for the construction of higher-order approximations. The coupling method is adopted for that purpose, which aims at finding a random variable whose law is easy to generate and is close to the target distribution. The last topic is motivated by the simulation of equations driven by Lévy processes, for which the main difficulty is to generalise some coupling results for the one-dimensional central limit theorem to the multi-dimensional case.

Este texto apresenta alguns dos elementos básicos envolvidos em um estudo introdutório das equações diferencias estocásticas. Tais equações modelam problemas a tempo contínuo em que as grandezas de interesse estão sujeitas a certos tipos de perturbações aleatórias. Em nosso estudo, a aleatoriedade nessas equações será representada por um termo que envolve o processo estocástico conhecido como Movimento Browniano. Para um tratamento matematicamente rigoroso dessas equações, faremos uso da Integral Estocástica de Itô. A construção dessa integral é um dos principais objetivos do texto. Depois de desenvolver os conceitos necessários, apresentaremos alguns exemplos e provaremos existência e unicidade de solução para equações diferenciais estocásticas satisfazendo certas hipóteses. / This text presents some of the basic elements involved in an introductory study of stochastic differential equations. Such equations describe certain kinds of random perturbations on continuous time models. In our study, the randomness in these equations will be represented by a term involving the stochastic process known as Brownian Motion. For a mathematically rigorous treatment of these equations, we use the Itô Stochastic Integral. The construction of this integral is one of the main goals of the text. After developing the necessary concepts, we present some examples and prove existence and uniqueness of solution of stochastic differential equations satisfying some hypothesis.

Este texto apresenta alguns dos elementos básicos envolvidos em um estudo introdutório das equações diferencias estocásticas. Tais equações modelam problemas a tempo contínuo em que as grandezas de interesse estão sujeitas a certos tipos de perturbações aleatórias. Em nosso estudo, a aleatoriedade nessas equações será representada por um termo que envolve o processo estocástico conhecido como Movimento Browniano. Para um tratamento matematicamente rigoroso dessas equações, faremos uso da Integral Estocástica de Itô. A construção dessa integral é um dos principais objetivos do texto. Depois de desenvolver os conceitos necessários, apresentaremos alguns exemplos e provaremos existência e unicidade de solução para equações diferenciais estocásticas satisfazendo certas hipóteses. / This text presents some of the basic elements involved in an introductory study of stochastic differential equations. Such equations describe certain kinds of random perturbations on continuous time models. In our study, the randomness in these equations will be represented by a term involving the stochastic process known as Brownian Motion. For a mathematically rigorous treatment of these equations, we use the Itô Stochastic Integral. The construction of this integral is one of the main goals of the text. After developing the necessary concepts, we present some examples and prove existence and uniqueness of solution of stochastic differential equations satisfying some hypothesis.

We study a new method for the strong approximate solution of stochastic differential equations using coupling and we prove order one error bounds for the new scheme in Lp space assuming the invertibility of the diffusion matrix. We introduce and implement two couplings called the exact and approximate coupling for this scheme obtaining good agreement with the theoretical bound. Also we describe a method for non-invertibility case (Combined method) and we investigate its convergence order which will give O(h3/4 √log(h)j) under some conditions. Moreover we compare the computational results for the combined method with its theoretical error bound and we have obtained a good agreement between them. In the last part of this thesis we work out the performance of the multilevel Monte Carlo method using the new scheme with the exact coupling and we compare the results with the trivial coupling for the same scheme.

Orientador: Maria Cristina de Castro Cunha / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-09T14:47:39Z (GMT). No. of bitstreams: 1 Dorini_FabioAntonio_D.pdf: 1226170 bytes, checksum: e29fb88b09843fe42235d804cbd8b789 (MD5) Previous issue date: 2007 / Resumo: Modelos matemáticos para processos do mundo real freqüentemente têm a forma de sistemas de equações diferenciais parciais. Estes modelos usualmente envolvem parâmetros como, por exemplo, os coeficientes no operador diferencial, e as condições iniciais e de fronteira. Tipicamente, assume-se que os parâmetros são conhecidos, ou seja, os modelos são considerados determinísticos. Entretanto, em situações mais reais esta hipótese freqüentemente não se verifica dado que a maioria dos parâmetros do modelo possui uma característica aleatória ou estocástica. Modelos avançados costumam levar em consideração esta natureza estocástica dos parâmetros. Em vista disso, certos componentes do sistema são modelados como variáveis aleatórias ou funções aleatórias. Equações diferenciais com parâmetros aleatórios são chamadas equações diferenciais aleatórias (ou estocásticas). Novas metodologias matemáticas têm sido desenvolvidas para lidar com equações diferenciais aleatórias, entretanto, este problema continua sendo objeto de estudo de muitos pesquisadores. Assim sendo, é importante a busca por novas formas (numéricas ou analíticas) de tratar equações diferenciais aleatórias. Durante a realização do curso de doutorado, vislumbrando a possibilidade de aplicações futuras em problemas de fluxo de fluidos em meios porosos (dispersão de poluentes e fluxos bifásicos, por exemplo), desenvolvemos trabalhos relacionados à equação do transporte linear unidimensional aleatória e ao problema de Burgers-Riemann unidimensional aleatório. Nesta tese, apresentamos uma nova metodologia, baseada nas idéias de Godunov, para tratar a equação do transporte linear unidimensional aleatória e desenvolvemos um eficiente método numérico para os momentos estatísticos da equação de Burgers-Riemann unidimensional aleatória. Para finalizar, apresentamos também novos resultados para o caso multidimensional: mostramos que algumas metodologias propostas para aproximar a média estatística da solução da equação do transporte linear multidimensional aleatória podem ser válidas para todos os momentos estatísticos da solução / Abstract: Mathematical models for real-world processes often take the form of systems of artial differential equations. Such models usually involve certain parameters, for example, the coefficients in the differential operator, and the initial and boundary conditions. Usually, all the model parameters are assumed to be known exactly. However, in realistic situations many of the parameters may have a random or stochastic character. More advanced models must take this stochastic nature into account. In this case, the components of the system are then modeled as random variables or random fields. Differential equations with random parameters are called random (or stochastic) differential equations. New mathematical methods have been developed to deal with this kind of problem, however, solving this problem is still the goal of several researchers. Thus, it is important to look for new approaches (numerical or analytical) to deal with random differential equations. Throughout the realization of the doctorate and looking toward future applications in porous media flow (pollution dispersal and two phase flows, for instance) we developed works related to the one-dimensional random linear transport equation and to the onedimensional random Burgers-Riemann problem. In this thesis, based on Godunov¿s ideas, we present a new methodology to deal with the one-dimensional random linear transport equation, and develop an efficient numerical scheme for the statistical moments of the solution of the one-dimensional random Burgers-Riemann problem. Finally, we also present new results for the multidimensional case: we have shown that some approaches to approximate the mean of the solution of the multidimensional random linear transport equation may be valid for all statistical moments of the solution / Doutorado / Analise Numerica / Doutor em Matemática Aplicada