La donnée d'une structure de Poisson sur une variété induit un feuilletage dont les feuilles sont des variétés symplectiques. En chaque point de la variété, il existe un unique invariant local, donné par la structure transverse. Dans ce travail, on s'intéresse aux invariants de nature emi-locale, c'est à dire associés au germe de la structure le ong d'une sous-variété. On s'interesse à deux cas extrêmes: celui d'une sous-variété de singularités sous des hypothèses génériqus, ainsi que celui du voisinage d'une feuille symplectique.

Many queueing systems have an arrival process that can be modeled by a Markov-modulated Poisson process. The Markov-modulated Poisson process (MMPP) is a doubly stochastic Poisson process in which the arrival rate varies according to a finite state irreducible Markov process. In many applications of MMPPs, the point process is constructed by superpositions or similar constructions, which lead to modulating Markov processes with a large state space. Since this limits the feasibility of numerical computations, a useful problem is to approximate an MMPP represented by a large Markov process by one with fewer states. We focus our attention in particular, to approximating a simple but useful special case of the MMPP, namely the Birth and Death Modulated Poisson process. In the validation stage, the quality of the approximation is examined in relation to the MMPP/G/1 queue.

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published_or_final_version / Applied Statistics / Master / Master of Social Sciences

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Ginzburg and Weinstein proved in [GW92] that for a compact, semisimple Lie group K endowed with the Lu-Weinstein Poisson structure, there exists a Poisson diffeomorphism from the dual Poisson Lie group K* to the dual k* of the Lie algebra of K endowed with the Lie-Poisson structure. We investigate the possibility of extending this result to the pseudo-unitary groups SU (p, q ), which are semisimple but not compact. The main results presented here are the following. (1) The Ginzburg-Weinstein proof hinges on the existence of a certain vector field X on k*. We prove that for any p, q, the analogous vector field for the SU (p, q ) case exists on an open subset of k*. (2) Each generic dressing orbit ψ(λ) in the Poisson dual AN can be embedded in the complex flag manifold K/T . We show that for SU (1, 1) and SU (1, 2), the induced Poisson structure π(λ) on ψ(λ) extends smoothly to the entire flag manifold. (3) Finally, we prove the Ginzburg-Weinstein theorem for the SU (1, 1) case in two different ways: first, by constructing the vector field X in coordinates and proving that it satisfies the necessary properties, and second, by adapting the approach of [FR96] to the SU (1, 1) case.

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Generated by stochastic recursions, perpetuities encompass a vast range of discretetime financial behaviours. When focusing on the dramatic changes occurring in such processes, the analysis of threshold exceedances provides an extensive description of their underlying mechanisms. Asymptotically, an exceedance point process tends to a compound Poisson measure, highlighting a tendency to cluster. Now, the parameters of this limit law are known, but complex. Here, an empirical approach is adopted, and a class of explicit compound Poisson models developed, with a bound on the error, for the exceedance point process of a finite, multidimensional perpetuity. In a financial regulatory context, this provides a new way of examining the Value-at-Risk criterion for securities.

Thesis (M.S.)--Virginia Polytechnic Institute and State University, 1994. / Vita. Abstract. Includes bibliographical references (leaves 127-128). Also available via the Internet.