We consider an SDE driven by a Lévy noise on a foliated manifold, whose trajectories stay on compact leaves. We determine the effective behavior of the system subject to a small smooth transversal perturbation of positive order epsilon. More precisely, we show that the average of the transversal component of the SDE converges to the solution of a deterministic ODE, according to the average of the perturbing vector field with respect to the invariant measures on the leaves (of the unpertubed system) as epsilon goes to 0. In particular we give upper bounds for the rates of convergence. The main results which are proved for pure jump Lévy processes complement the result by Gargate and Ruffino for Stratonovich SDEs to Lévy driven SDEs of Marcus type.

Orientador: Paulo Régis Caron Ruffino / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-23T11:21:58Z (GMT). No. of bitstreams: 1 Costa_PauloHenriquePereirada_D.pdf: 1382380 bytes, checksum: 975aac3916932e92b8fe92b185b6eb9f (MD5) Previous issue date: 2013 / Resumo: Seja (M,?) uma variedade Riemanniana compacta folheada. Consideramos uma família de semigrupos Feller compatível em C(Mn) associada as leis de um processo Markoviano de n-pontos. Com algumas condições (Le Jan e Raimond [34]) existe um fluxo estocástico de aplicações mensuráveis em M. Estudamos aqui a degenerescência desses semigrupos tais que o fluxo de aplicações seja folheado, ou seja, cada trajetória permanece na folha em que começou q.s. e portanto cria uma obstrução geométrica natural para a coalescência de trajetórias em folhas distintas. Como uma aplicação dessa teoria, um princípio de médias é provado para uma perturbação de primeira ordem transversal as folhas. Estimativas de taxas de convergências também são dadas / Abstract: Let (M,?) be a compact Riemannian foliated manifold. We consider a family of compatible Feller semigroups in C(Mn) associated to laws of the n-point motion. Under some assumptions (Le Jan and Raimond [34]) there exists a stochastic flow of measurable mappings in M. We study the degeneracy of these semigroups such that the flow of mappings is foliated, i.e. each trajectory lays in a single leaf of the foliation a.s, hence creating a geometrical obstruction for coalescence of trajectories in different leaves. As an application, an averaging principle is proved for a first order perturbation transversal to the leaves. Estimates for the rate of convergence are calculated / Doutorado / Matematica / Doutor em Matemática

Orientador: Paulo Regis Caron Ruffino / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T22:03:50Z (GMT). No. of bitstreams: 1 GonzalesGargate_IvanItalo_D.pdf: 724522 bytes, checksum: bb313a00b360e7bea2a2411b759c7389 (MD5) Previous issue date: 2012 / Resumo: Nesta tese, estudamos um princípio de médias em equações diferenciais estocásticas sobre variedades folheadas com folhas compactas. Começaremos introduzindo o princípio de médias sobre equações diferenciais ordinárias reais. A título de comparação vamos rever conceitos básicos de variedade simplética com a finalidade de comparar/estender os resultados obtidos por Xue-Mei Li sobre um princípio de médias para um sistema Hamiltoniano estocástico completamente integrável. Nosso principal resultado é generalizar estas idéias para o caso de uma variedade M = (-a; a)n x N, onde N é uma variedade compacta sem bordo. Em particular mostraremos nossos resultados para o caso que a folheação é gerada por uma submersão de M sobre Rn. Finalmente apresentamos alguns exemplos / Abstract: In this thesis, we study the averaging principle for stochastic differential equations on foliated manifolds with compact leaves. We begin by introducing the averaging principle over real ordinary differential equations. For comparison we will review basic concepts of symplectic manifold in order to compare/extend the results obtained by Xue-Mei Li about a averaging principle for a completely integrable stochastic Hamiltonian system. Our main result is to generalize these ideas to the case of a manifold M = (-a; a)n x N, where N is a compact manifold without boundary. In particular our results show for the case that foliation is generated by an submersion of M over Rn. Finally we present some examples / Doutorado / Matematica / Doutor em Matemática

Im ersten Teil der Dissertation wird ein diskretes stochastisches zustandsabhängiges Modell eines zweiseitigen Limit Orderbuchs als bestehend aus den Zustandsgrößen bester Bidpreis (Geldkurs), bester Askpreis (Briefkurs) und vorhandener Kauf- bzw. Verkaufsdichte definiert. Für eine einfache Skalierung mit zwei Zeitskalen wird ein Grenzwertsatz bewiesen. Die Veränderungen der besten Bid- und Askpreise werden im Sinne des Gesetzes der großen Zahlen skaliert und dies entspricht der langsameren Zeitskala. Das Platzieren bzw. Stornieren der Limitorder findet auf der schnelleren Zeitskala statt. Der Grenzwertsatz besagt, dass die fundamentalen Zustandsgrößen, gegeben Regularitätsbedingungen der einkommenden Order, fast sicher zu einem stetigen Limesmodell konvergieren. Im Limesmodell sind der beste Bidpreis und der beste Askpreis die eindeutigen Lösungen von zwei gekoppelten gewöhnlichen DGLen. Die Kauf- und Verkaufsdichten sind jeweils als eindeutige Lösungen von linearen hyperbolischen PDGLen, die anhand der Erwartungswerte der einkommenden Orderparameter festgelegt sind, gegeben. Die Lösungen sind in geschlossener Form erhältlich. Im zweiten Teil wird ein funktionaler zentraler Grenzwertsatz d.h. ein Invarianzprinzip für ein vereinfachtes Modell eines Limitorderbuches bewiesen. Unter einer natürlichen Skalierung konvergiert der zweidimensionale Preisprozess (Bid- und Askpreis) in Verteilung zu einer Semimartingal reflektierten Brownschen Bewegung in der zugelassenen Preismenge. Gleichzeitig konvergieren die Kauf- und Verkaufsdichten im schwachen Sinn zum Betrag einer zweiparametrischen Brownschen Bewegung. Es wird weiterhin anhand eines Beispiels gezeigt, wie man für das Modell im ersten Teil eine stochastiche PDGL, unter einer starken Stationaritätsannahme für die Orderplatzierungen und -stornierungen, herleiten kann. Im dritten Teil wird ein Mittelungs- bzw. ein Invarianzprinzip für diskrete Banach- bzw. Hilbertraumwertige stochastische Prozesse bewiesen. / In the first part of the thesis, we define a random state-dependent discrete model of a two-sided limit order book in terms of its key quantities best bid [ask] price and the standing buy [sell] volume density. For a simple scaling that introduces a slow time scaling, that is equivalent to the classical law of large numbers, for the bid/ask prices and a faster time scale for the limit volume placements/cancelations, that keeps the expected volume rate over the considered price interval invariant, we prove a limit theorem. The limit theorem states that, given regularity conditions on the random order flow, the key quantities converge in the sense of a strong law of large numbers to a tractable continuous limiting model. The limiting model is such that the best bid and ask price dynamics can be described in terms of two coupled ODE:s, while the dynamics of the relative buy and sell volume density functions are given as the unique solutions of two linear first-order hyperbolic PDE:s with variable coefficients, specified by the expectation of the order flow parameters. In the second part, we prove a functional central limit theorem i.e. an invariance principle for an order book model with block shaped volume densities close to the spread. The weak limit of the two-dimensional price process (best bid and ask price) is given by a semi-martingale reflecting Brownian motion in the set of admissible prices. Simultaneously, the relative buy and sell volume densities close to the spread converge weakly to the modulus of a two-parameter Brownian motion. We also demonstrate an example how to easily derive an SPDE for the relative volume densities in a simple case, when a strong stationarity assumption is made on the limit order placements and cancelations for the model suggested in the first part. In the third and final part of the thesis, we prove an averaging and an invariance principle for discrete processes taking values in Banach and Hilbert spaces, respectively.