Title: Stochastic Catastrophe Model Cusp Author: Jan Voříšek Department: Department of Probability and Mathematical Statistics Supervisor: Prof. Ing. Miloslav Vošvrda, CSc., Czech Academy of Sciences, Institute of Information Theory and Automation Abstract: The goal of this thesis is to analyze the stochastic cusp model. This task is divided into two main topics. The first of them concentrates on the stationary density of the cusp model and statistical testing of its bimodality, where power and size of the proposed tests are simulated and compared with the dip test of unimodality. The second main topic deals with the transition density of the stochastic cusp model. Comparison of approximate maximum likelihood approach with traditional finite difference and numerical simulations indicates its advantage in terms of speed of estimation. An approximate Fisher information matrix of general stochastic process is derived. An application of the cusp model to the exchange rate with time-varying parameters is estimated, the extension of the cusp model into stochastic bimodality model is proposed, and the measure of probability of intrinsic crash of the cusp model is suggested. Keywords: stochastic cusp model, bimodality testing, transition density ap- proximation

In this thesis, we provide convergent numerical solutions to non-linear forward-BSDEs (Backward Stochastic Differential Equations). Applications in mathematical finance, financial economics and financial econometrics are discussed. Numerical examples show the effectiveness of our methods.

<p>This thesis is centered around three infinite dimensional diffusion processes:</p> <p>(i). the infinitely-many-neutral-alleles diffusion model [Ethier and Kurtz, 1981],</p> <p>(ii). the two-parameter infinite dimensional diffusion model [Petrov, 2009] and [Feng and Sun, 2010],</p> <p>(iii). the infinitely-many-alleles diffusion with symmetric dominance [Ethier and Kurtz, 1998].</p> <p>The partition structures, the ergodic inequalities and the asymptotic theory of these three models are discussed. In particular, the asymptotic theory turns out to be the major contribution of this thesis.</p> <p>In Chapter 2, a slightly altered version of Kingman's one-to-one correspondence theorem on partition structures is provided, which in turn becomes a handy tool for obtaining the asymptotic result on the partition structures associated with models (i) and (ii).</p> <p>In Chapter 3, the three diffusion models are briefly introduced. New representations of the transition densities of models (i) and (ii) are obtained simply by rearranging the previous representations obtained in [Ethier, 1992] and [Feng et al., 2011] respectively. These two new representations have their own advantages, by making use of which the corresponding ergodic inequalities easily follow. Furthermore, thanks to the functional inequalities in [Feng et al., 2011], the ergodic inequality for model (iii) becomes available as well.</p> <p>In Chapter 4, the asymptotic properties of models (i) and (ii) are thoroughly studied. Various asymptotic results are obtained, such as the weak limits of models (i) and (ii) at different time scales when the mutation rate approaches infinity, and the large deviation principle for models (i) and (ii) at a fixed time, and that of the transient partition structures of models (i) and (ii). Of all these results, the weak limit and the large deviation principle of the transient partition structures are of particular interest.</p> <p>In Chapter 5, the asymptotic results on the stationary distribution and the transient distribution of model (iii) are both obtained. The weak limit of the infinitely-many- alleles diffusion with symmetric overdominance at fixed time t serves as an alternative answer to Gillespie's conjecture [Gillespie, 1999]. The weak limit of the stationary distribution of the infinitely-many-alleles diffusion with symmetric overdominance provides a complete solution to the remaining problem in [Feng, 2009].</p> / Doctor of Philosophy (PhD)

In this paper I derive the asymptotics of the exact, Euler, and Milstein ML estimators for diffusion models, including general nonstationary diffusions. Though there have been many estimators for the diffusion model, their asymptotic properties were generally unknown. This is especially true for the nonstationary processes, even though they are usually far from the standard ones. Using a new asymptotics with respect to both the time span T and the sampling interval ¢, I find the asymptotics of the estimators and also derive the conditions for the consistency. With this new asymptotic result, I could show that this result can explain the properties of the estimators more correctly than the existing asymptotics with respect only to the sample size n. I also show that there are many possibilities to get a better estimator utilizing this asymptotic result with a couple of examples, and in the second part of the paper, I derive the higher order asymptotics which can be used in the bootstrap analysis.

Made available in DSpace on 2015-03-03T15:28:31Z (GMT). No. of bitstreams: 1 EnaiTC_DISSERT.pdf: 791005 bytes, checksum: 0aee506aa23dd64244f916f2b42bdf10 (MD5) Previous issue date: 2010-05-21 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / In this work, we studied the strong consistency for a class of estimates for a transition density of a Markov chain with general state space ?E ? Rd. The strong ergodicity of the estimates for the density transition is obtained from the strong consistency of the kernel estimates for both the marginal density p(:) of the chain and the joint density q(., .). In this work the Markov chain is supposed to be homogeneous, uniformly ergodic and possessing a stationary density p(.,.) / No presente trabalho, estudamos a consist?ncia forte de uma classe de estimadores do tipo n?cleo para a densidade de transi??o de uma cadeia de Markov com espa?o de estados geral. A consist?ncia forte do estimador da densidade de transi??o, foi obtida a partir da consist?ncia forte dos estimadores do tipo n?cleo para a densidade marginal da cadeia, , e da consist?ncia forte dos estimadores do tipo n?cleo para a densidade conjunta, . Foi considerado que a cadeia ? homog?nea e uniformemente erg?dica com densidade inicial estacion?ria