The main topic of this dissertation is the estimation of the Hurst index H of the solutions of stochastic differential equations (SDEs) driven by the fractional Brownian motion (fBm). Firstly, the limit behavior of the first and second order quadratic variations of the solutions of SDEs driven by the fBm is analyzed. This yields several strongly consistent estimators of the Hurst index H. Secondly, it is proved that in case the solution of the SDE is replaced by its Milstein approximation, the estimators remain strongly consistent. Additionally, the possibilities of applying the increment ratios (IR) statistic based estimator of H originally obtained by J. M. Bardet and D. Surgailis in 2010 to the fractional geometric Brownian motion are examined. Furthermore, this dissertation derives the convergence rate of the modified Gladyshev’s estimator of the Hurst index to its real value. The estimators obtained in the dissertation were compared with several other known estimators of the Hurst index H, namely the naive and ordinary least squares Gladyshev and eta-summing oscillation estimators, the variogram estimator and the IR estimator. The models chosen for comparison of these estimators were the fractional Ornstein-Uhlenbeck (O-U) process and the fractional geometric Brownian motion (gBm). The initial inference about the behavior of these estimators was drawn for the O-U process which is Gaussian, while the gBm process was used to check how the estimators behave in a... [to full text] / Pagrindinė šios disertacijos tema – stochastinių diferencialinių lygčių (SDL), valdomų trupmeninio Brauno judesio (tBj), sprendinių Hursto indekso H vertinimas. Pirmiausia disertacijoje išnagrinėta SDL, valdomų tBj, sprendinių pirmos ir antros eilės kvadratinių variacijų ribinė elgsena. Iš šių rezultatų seka keli stipriai pagrįsti Hursto indekso H įvertiniai. Įrodyta, kad šie įvertiniai išlieka stipriai pagrįsti, jei tikra sprendinio trajektorija keičiama jos Milšteino aproksimacija. Taip pat išnagrinėtos pokyčių santykio (increment ratios) statistikos H įvertinio, gauto J. M. Bardeto ir D. Surgailio 2010 m., taikymo trupmeninio geometrinio Brauno judesio Hursto indekso vertinimui galimybės bei nustatytas modifikuoto Gladyševo H įvertinio konvergavimo i tikrąją parametro reikšme greitis. Gauti įvertiniai palyginti su kai kuriais kitais žinomais Hursto indekso H įvertiniais: naiviais bei mažiausių kvadratų Gladyševo ir eta-sumavimo osciliacijos įvertiniais, variogramos įvertiniu ir pokyčių santykio statistikos įvertiniu. Įvertinių elgsena buvo palyginta trupmeniniam Ornšteino-Ulenbeko (OU) procesui bei trupmeniniam geometriniam Brauno judesiui (gBj). Pradinės išvados buvo padarytos O-U procesui, kuris yra Gauso, o gBj procesas buvo naudojamas patikrinti, kaip šie įvertiniai elgiasi, kai procesas yra ne Gauso. Disertaciją sudaro įvadas, 3 pagrindiniai skyriai, išvados, literatūros sąrašas, autoriaus publikacijų disertacijos tema sąrašas ir du priedai.

We propose a simple and general method to obtain the boundary crossing probability for Brownian motion. This method can be easily extended to higher dimensional of Brownian motion. It also covers certain classes of stochastic processes associated with Brownian motion. The basic idea of the method is based on being able to construct a nite Markov chain such that the boundary crossing probability of Brownian motion is obtained as the limiting probability of the nite Markov chain entering a set of absorbing states induced by the boundary. Numerical results are given to illustrate our method.

In this thesis Langevin approach is applied to analyze the nuclear dissipative dynamics in fission and fusion reactions. In these investigations, the nuclear elongation coordinate and the corresponding momentum are chosen as collective variables. By considering changes in these variables the decay rate of fission and the formation probability of fusion for heavy ion reactions are calculated. These calculations are performed using simulation techniques and the results thus obtained are compared with the corresponding results of analytic solutions.

[pt] Esta dissertação procura analisar através de um estudo de caso, as alternativas de desenvolvimento de um campo de petróleo já descoberto, mas ainda não explorado, utilizando a Teoria das Opções Reais. A partir deste estudo, será possível avaliar uma alternativa de desenvolvimento da produção de dois poços de petróleo, que serão explorados no futuro, dependendo das condições de mercado e das informações técnicas geradas pela produção inicial do campo. A dissertação tem como principal objetivo comparar os resultados das incertezas de mercado no preço do petróleo representadas pelos processos estocásticos, o Movimento Geométrico Browniano e o Processo de Reversão à Média com Saltos, para determinação da ferramenta gerencial denominada Curva de Gatilho. / [en] This dissertation search for to analyze through a study of case, the alternatives of development of a oil field already discovered, but not yet exploited, using the Theory of the Real Options. From this study, it will be possible to evaluate an alternative of development of the production of two wells, that will be explored in the future, depending on the market conditions and of the technical informations generated for the initial production of the field. The dissertation has as mean objective to compare the results of the uncertainties of market in the price of oil rerepresented by Estocastic Processes, the Geometric Browniano Movement and the Process of Mean Reversion with Jumps, for determination of the management tool named of Trigger.

Title: Stochastic evolution equations with multiplicative fractional noise Author: Jana Šnupárková Departement: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Bohdan Maslowski, DrSc. Supervisor's e-mail address: maslow@karlin.mff.cuni.cz Abstract: The fractional Gaussian noise is a formal derivative of a fractional Brownian motion with Hurst parameter H ∈ (0, 1). An explicit formula for a solution to stochastic differential equations with a multiplicative fractional Gaussian noise in a separable Hilbert space is given. The large time behaviour of the solution is studied. In addition, equations of this type with a nonlinear perturbation of a drift part are investigated in the case H > 1/2. Keywords: Fractional Brownian Motion, Stochastic Differential Equations in Hilbert Space, Explicit Formula for Solution

Stochastic Evolution Equations Petr Čoupek Doctoral Thesis Abstract Linear stochastic evolution equations with additive regular Volterra noise are studied in the thesis. Regular Volterra processes need not be Gaussian, Markov or semimartingales, but they admit a certain covariance structure instead. Particular examples cover the fractional Brownian motion of H > 1/2 and, in the non-Gaussian case, the Rosenblatt process. The solution is considered in the mild form, which is given by the variation of constants formula, and takes values either in a separable Hilbert space or the space Lp(D, µ) for large p. In the Hilbert-space setting, existence, space-time regularity and large-time behaviour of the solutions are studied. In the Lp setting, existence and regularity is studied, and in concrete cases of stochastic partial differential equations, the solution is shown to be a space-time continuous random field.

>Magister Scientiae - MSc / We present various methods of pricing Asian options. The methods include Monte Carlo simulations designed using control and antithetic variates, numerical solution of partial differential equation and using lower bounds.The price of the Asian option is known to be a certain risk-neutral expectation. Using the Feynman-Kac theorem, we deduce that the problem of determining the expectation implies solving a linear parabolic partial differential equation. This partial differential equation does not admit explicit solutions due to the fact that the distribution of a sum of lognormal variables is not explicit. We then solve the partial differential equation numerically using finite difference and Monte Carlo methods.Our Monte Carlo approach is based on the pseudo random numbers and not deterministic sequence of numbers on which Quasi-Monte Carlo methods are designed. To make the Monte Carlo method more effective, two variance reduction techniques are discussed.Under the finite difference method, we consider explicit and the Crank-Nicholson’s schemes. We demonstrate that the explicit method gives rise to extraneous solutions because the stability conditions are difficult to satisfy. On the other hand, the Crank-Nicholson method is unconditionally stable and provides correct solutions. Finally, we apply the pricing methods to a similar problem of determining the price of a European-style arithmetic basket option under the Black-Scholes framework. We find the optimal lower bound, calculate it numerically and compare this with those obtained by the Monte Carlo and Moment Matching methods.Our presentation here includes some of the most recent advances on Asian options, and we contribute in particular by adding detail to the proofs and explanations. We also contribute some novel numerical methods. Most significantly, we include an original contribution on the use of very sharp lower bounds towards pricing European basket options.

Nous avons tou(te)s été témoins des nuages d'étourneaux dans le ciel ou de la formation de bancs de poissons dans l'océan. Ce type d'organisation chez les êtres vivant se produit aussi à des échelles parfois invisibles pour l'oeil humain: celles des micro-organismes. Les suspensions de micro-nageurs présentent une dynamique riche. Elles peuvent former des structures cohérentes résultant d'un mouvement collectif, mélanger le fluides environnant et/ou modifier ses propriétés rhéologiques. Leurs comportements peuvent jouer un rôle important dans la survie, l'équilibre des espèces, leur stratégie trophique et même pour la fertilité animale. La diversité des phénomènes observés résulte de l'interaction complexe entre mécanismes de nage, processus physiologiques, processus chimiques et interactions hydrodynamiques. Comprendre et maîtriser les mécanismes impliqués fait nécessairement appel la Mécanique des Fluides. Les études expérimentales permettent de mettre en exergue certains phénomènes et parfois de les expliquer. Cependant la modélisation s'avère indispensable. Or, inclure une description fine des mécanismes de nages dans une suspension contenant des milliers (voire des millions) d'individus, implique de considérer une vaste gamme d'échelles couplées (typiquement du micron 10^-6m au millimètre 10^-3m). Décrire une physique multi-échelles pour ce type problème reste un défi majeur pour la modélisation numérique actuelle. Ainsi, dans le cadre de cette thèse nous nous proposons d'apporter une contribution dans cette direction. Nous montrerons dans une premiere partie qu'il est possible de reproduire les mécanismes de nage de façon satisfaisante à l'échelle du micro-organisme avec des modèles de différentes complexités. Nous présenterons ensuite nos développements pour étendre ces modèles a l'échelle de la suspension. Nous montrerons comment inclure simultanément les effets Browniens qui agissent sur les plus petite particules (10^-6m). Enfin, nous exploiterons l'outil mis en place pour simuler des suspensions actives. Sa capacité à reproduire certains résultats de la littérature à précision égale, à moindre coût et à plus grande échelle, permet de combler le fossé entre modèles individuels, travaux expérimentaux et modèles continus issus de la théorie cinétique. Forts de cet outil, nous tenterons de répondre à deux questions ouvertes dans la littérature expérimentale : l'origine des corrélations d'orientation dans les suspensions de microgouttes auto-propulsées et les mécanismes en jeu dans la diffusion des particules Browniennes dans les suspensions actives. / We have all witnessed the flocking of starlings in the sky and the schools of fish that form in the ocean. This kind of organization of living creatures is not limited to those that we see, but also occurs for those that we don’t : swimming microorganisms. Suspen- sions of micro-swimmers exhibit a rich dynamics. Their behaviors can play an important role in the survival of the group, its development, the balance between species, their trophic strategies and even animal fertility. They can form coherent structures due to collective motion, mix the surrounding fluid or modify its rheological properties. Such diversity results from the complex interplay between swimming strategies, physiological processes, chemical reactions and hydrodynamic interactions. Fluid Mechanics is there- fore essential to understand and master the mechanisms involved in these phenomena. While experimental studies bring out new findings and, sometimes, provide physical ex- planations, modeling remains essential. Yet, including an accurate description of the micro-swimmers in a suspension containing thousands (nay millions) individuals, requires considering a wide range of coupled scales (from one micron 10^−6m to several millimeters 10^−3m). What happens on large scales depends on sophisticated mechanisms occurring two or three orders of magnitude below. Therefore, the multiscale modeling of such phenomena is still a major challenge for the state-of-the-art numerical methods. This thesis aims at providing a contribution in that direction. In a first part, we will show that reproducing swimming mechanisms at the scale of the micro-swimmer can be achieved with various models spanning different levels of complexity. We will then present our developments to incorporate these models in an efficient framework for large scale simulations. We will show how to simultaneously account for the Brownian motion of the smallest particles (10^−6m). Our code reproduces known results from the literature with the same accuracy, but at lower cost and at larger scales, thus bridging a gap between particle-based models, experiments and continuum formulations from kinetic theory. Using the capabilities afforded by our method, we eventually address two open problems in the experimental literature : the origins of orientational correla- tions between interacting self-propelled micro-droplets and the mechanisms at play in the nonlinear enhancement of Brownian particle diffusion in active suspensions.

Cette thèse porte sur l'existence de champs browniens fractionnaires indexés par des variétés riemanniennes. Ces objets héritent des propriétés qui font le succès du mouvement brownien fractionnaire classique (H-autosimilarité des trajectoires ajustable, accroissements stationnaires), mais autorisent à considérer des applications où les données sont portées par un espace qui peut par exemple être courbé ou troué. L'existence de ces champs n'est assurée que lorsque la quantité 2H est inférieure à l'indice fractionnaire de la variété, qui n'est connu que dans un petit nombre d'exemples. Dans un premier temps nous donnons une condition nécessaire pour l'existence de champ brownien fractionnaire. Dans le cas du champ brownien (correspondant à H=1/2) indexé par des variétés qui ont des géodésiques fermées minimales, cette condition s'avère très contraignante : nous donnons des résultats de non-existence dans ce cadre, et montrons notamment qu'il n'existe pas de champ brownien indexé par une variété compacte non simplement connexe. La condition nécessaire donne également une preuve courte d'un fait attendu qui est la non-dégénérescence du champ brownien indexé par les espaces hyperboliques réels. Dans un second temps nous montrons que l'indice fractionnaire du cylindre est nul, ce qui constitue un exemple totalement dégénéré. Nous en déduisons que l'indice fractionnaire d'un espace métrique n'est pas continu par rapport à la convergence de Gromov-Hausdorff. Nous généralisons ce résultat sur le cylindre à un produit cartésien qui possède une géodésique fermée minimale, et donnons une majoration de l'indice fractionnaire de surfaces asymptotiquement proches du cylindre au voisinage d'une géodésique fermée minimale. / The aim of the thesis is the study of the existence of fractional Brownian fields indexed by Riemannian manifolds. Those fields inherit key properties of the classical fractional Brownian motion (sample paths with self-similarity of adjustable parameter H, stationary increments), while allowing to consider applications with data indexed by a space which can be for example curved or with a hole. The existence of those fields is only insured when the quantity 2H is inferior or equal to the fractional index of the manifold, which is known only in a few cases. In a first part we give a necessary condition for the fractional Brownian field to exist. In the case of the Brownian field (corresponding to H=1/2) indexed by a manifold with minimal closed geodesics this condition happens to be very restrictive. We give several nonexistence results in this situation. In particular we show that there exists no Brownian field indexed by a nonsimply connected compact manifold. Our necessary condition also gives a short proof of an expected result: we prove the nondegeneracy of fractional Brownian fields indexed by the real hyperbolic spaces. In a second part we show that the fractional index of the cylinder is null, which gives a totally degenerate case. We deduce from this result that the fractional index of a metric space is noncontinuous with respect to the Gromov-Hausdorff convergence. We generalise this result about the cylinder to a Cartesian product with a closed minimal geodesic. Furthermore we give a bound of the fractional index of surfaces asymptotically close to the cylinder in the neighbourhood of a closed minimal geodesic.

This thesis deals with the stochastic integral with respect to Gaussian processes, which can be expressed in the form Bt = t 0 K(t, s)dWs. Here W stands for a Brownian motion and K for a square integrable Volterra kernel. Such processes generalize fractional Brownian motion. Since these processes are not semimartin- gales, Itô calculus cannot be used and other methods must be employed to define the stochastic integral with respect to these proceses. Two ways are considered in this thesis. If both the integrand and the process B are regular enough, it is possible to define the integral in the pathwise sense as a generalization of Lebesgue-Stieltjes integral. The other method uses the methods of Malliavin cal- culus and defines the integral as an adjoint operator to the Malliavin derivative. As an application, the stochastic differential equation dSt = µStdt + σStdBt, which is used to model price of a stock, is solved. Implications of such a model are briefly discussed. 1