Limit theorems in the sense of weak convergence of probability measures for the Lerch zeta-function with algebraic irrational parameter are obtained. A theorem of mentioned type on the complex plane, a joint limit theorem for a collection of Lerch zeta-functions on the complex plane as well as a limit theorem in the space of analytic functions are proved. The theorems obtained characterize the asymptotic behaviour of the Lerch zeta-function and can be applied in the investigation of the universality of that function. / Yra gautos Lercho dzeta funkcijos su algebriniu iracionaliuoju parametru ribinės teoremos silpno tikimybinių matų konvergavimo prasme. Yra įrodyta minėto tipo teorema kompleksinėje plokštumoje, jungtinė ribinė teorema Lercho dzeta funkcijų rinkiniui kompleksinėje plokštumoje ir teorema analizinių funkcijų erdvėje. Įrodytos teoremos charakterizuoja Lercho dzeta funkcijų asimptotinį elgesį ir gali būti taikomos šios funkcijos universalumui tirti.

Yra gautos Lercho dzeta funkcijos su algebriniu iracionaliuoju parametru ribinės teoremos silpno tikimybinių matų konvergavimo prasme. Yra įrodyta minėto tipo teorema kompleksinėje plokštumoje, jungtinė ribinė teorema Lercho dzeta funkcijų rinkiniui kompleksinėje plokštumoje ir teorema analizinių funkcijų erdvėje. Įrodytos teoremos charakterizuoja Lercho dzeta funkcijų asimptotinį elgesį ir gali būti taikomos šios funkcijos universalumui tirti. / Limit theorems in the sense of weak convergence of probability measures for the Lerch zeta-function with algebraic irrational parameter are obtained. A theorem of mentioned type on the complex plane, a joint limit theorem for a collection of Lerch zeta-functions on the complex plane as well as a limit theorem in the space of analytic functions are proved. The theorems obtained characterize the asymptotic behaviour of the Lerch zeta-function and can be applied in the investigation of the universality of that function.

A standard problem in mathematical finance is the calculation of the price of some financial derivative such as various types of options. Since there exists analytical solutions in only a few cases it will often boil down to estimating the price with Monte Carlo simulation in conjunction with some numerical discretization scheme. The upside of using what we can call standard Monte Carlo is that it is relative straightforward to apply and can be used for a wide variety of problems. The downside is that it has a relatively slow convergence which means that the computational cost or complexity can be very large. However, this slow convergence can be improved upon by using Multilevel Monte Carlo instead of standard Monte Carlo. With this approach it is possible to reduce the computational complexity and cost of simulation considerably. The aim of this thesis is to introduce the reader to the Multilevel Monte Carlo method with applications to European and Asian call options in both the Black-Scholes-Merton (BSM) model and in the Heston model. To this end we first cover the necessary background material such as basic probability theory, estimators and some of their properties, the stochastic integral, stochastic processes and Ito’s theorem. We introduce stochastic differential equations and two numerical discretizations schemes, the Euler–Maruyama scheme and the Milstein scheme. We define strong and weak convergence and illustrate these concepts with examples. We also describe the standard Monte Carlo method and then the theory and implementation of Multilevel Monte Carlo. In the applications part we perform numerical experiments where we compare standard Monte Carlo to Multilevel Monte Carlo in conjunction with the Euler–Maruyama scheme and Milsteins scheme. In the case of a European call in the BSM model, using the Euler–Maruyama scheme, we achieved a cost O(ε-2(log ε)2) to reach the desired error in accordance with theory in comparison to the O(ε-3) cost for standard Monte Carlo. When using Milsteins scheme instead of the Euler–Maruyama scheme it was possible to reduce the cost in terms of the number of simulations needed to achieve the desired error even further. By using Milsteins scheme, a method with greater order of strong convergence than Euler–Maruyama, we achieved the O(ε-2) cost predicted by the complexity theorem compared to the standard Monte Carlo cost of order O(ε-3). In the final numerical experiment we applied the Multilevel Monte Carlo method together with the Euler–Maruyama scheme to an Asian call in the Heston model. In this case, where the coefficients of the Heston model do not satisfy a global Lipschitz condition, the study of strong or weak convergence is much harder. The numerical experiments suggested that the strong convergence was slightly slower compared to what was found in the case of a European call in the BSM model. Nevertheless, we still achieved substantial savings in computational cost compared to using standard Monte Carlo.

Cette thèse est consacrée à l’étude de deux problèmes différents : l’asymptotique suramortie de la dynamique de Langevin d’une part, et l’étude d’une technique de réduction de variance dans une méthode de Monte Carlo par une repondération optimale des échantillons, d’autre part. Dans le premier problème, on montre la convergence en distribution de processus de Langevin dans l’asymptotique sur-amortie. La preuve repose sur la méthode classique des “fonctions test perturbées”, qui est utilisée pour montrer la tension dans l’espace des chemins, puis pour identifier la limite comme solution d’un problème de martingale. L’originalité du résultat tient aux hypothèses très faibles faites sur la régularité de l’énergie potentielle. Dans le deuxième problème, nous concevons des méthodes de réduction de la variance pour l’estimation de Monte Carlo d’une espérance de type E[φ(X, Y )], lorsque la distribution de X est exactement connue. L’idée générale est de donner à chaque échantillon un poids, de sorte que la distribution empirique pondérée qui en résulterait une marginale par rapport à la variable X aussi proche que possible de sa cible. Nous prouvons plusieurs résultats théoriques sur la méthode, en identifiant des régimes où la réduction de la variance est garantie. Nous montrons l’efficacité de la méthode en pratique, par des tests numériques qui comparent diverses variantes de notre méthode avec la méthode naïve et des techniques de variable de contrôle. La méthode est également illustrée pour une simulation d’équation différentielle stochastique de Langevin / This dissertation is devoted to studying two different problems: the over-damped asymp- totics of Langevin dynamics and a new variance reduction technique based on an optimal reweighting of samples.In the first problem, the convergence in distribution of Langevin processes in the over- damped asymptotic is proven. The proof relies on the classical perturbed test function (or corrector) method, which is used (i) to show tightness in path space, and (ii) to identify the extracted limit with a martingale problem. The result holds assuming the continuity of the gradient of the potential energy, and a mild control of the initial kinetic energy. In the second problem, we devise methods of variance reduction for the Monte Carlo estimation of an expectation of the type E [φ(X, Y )], when the distribution of X is exactly known. The key general idea is to give each individual sample a weight, so that the resulting weighted empirical distribution has a marginal with respect to the variable X as close as possible to its target. We prove several theoretical results on the method, identifying settings where the variance reduction is guaranteed, and also illustrate the use of the weighting method in Langevin stochastic differential equation. We perform numerical tests comparing the methods and demonstrating their efficiency

Le sujet principal de ce mémoire est l'étude de la distribution asymptotique de la fonction f_m qui compte le nombre de diviseurs premiers distincts parmi les nombres premiers $p_1,...,p_m$. Au premier chapitre, nous présentons les sept résultats qui seront démontrés au chapitre 4. Parmi ceux-ci figurent l'analogue du théorème d'Erdos-Kac et un résultat sur les grandes déviations. Au second chapitre, nous définissons les espaces de probabilités qui serviront à calculer les probabilités asymptotiques des événements considérés, et éventuellement à calculer les densités qui leur correspondent. Le troisième chapitre est la partie centrale du mémoire. On y définit la promenade aléatoire qui, une fois normalisée, convergera vers le mouvement brownien. De là, découleront les résultats qui formeront la base des démonstrations de ceux chapitre 1. / The main topic of this masters thesis is the study of the asymptotic distribution of the fonction f_m which counts the number of distinct prime divisors among the first $m$ prime numbers, i.e. $p_1,...,p_m$. The first chapter provides the seven main results which will later on be proved in chapter 4. Among these we find the analogue of the Erdos-Kac central limit theorem and a result on large deviations. In the following chapter, we define several probability spaces on which we will calculate asymptotic probabilities of specific events. These will become necessary for calculating their corresponding densities. The third chapter is the main part of this masters thesis. In it, we introduce a random walk which, when suitably normalized, will converge to the Brownian motion. We will then obtain results which will form the basis of the proofs of those of chapiter 1.

<p>This thesis, consisting of three papers and a summary, studies topics in the theory of stochastic processes related to long-range dependence. Much recent interest in such probabilistic models has its origin in measurements of Internet traffic data, where typical characteristics of long memory have been observed. As a macroscopic feature, long-range dependence can be mathematically studied using certain scaling limit theorems. </p><p>Using such limit results, two different scaling regimes for Internet traffic models have been identified earlier. In one of these regimes traffic at large scales can be approximated by long-range dependent Gaussian or stable processes, while in the other regime the rescaled traffic fluctuates according to stable ``memoryless'' processes with independent increments. In Paper I a similar limit result is proved for a third scaling scheme, emerging as an intermediate case of the other two. The limit process here turns out to be a non-Gaussian and non-stable process with long-range dependence.</p><p>In Paper II we derive a representation for the latter limit process as a stochastic integral of a deterministic function with respect to a certain compensated Poisson random measure. This representation enables us to study some further properties of the process. In particular, we prove that the process at small scales behaves like a Gaussian process with long-range dependence, while at large scales it is close to a stable process with independent increments. Hence, the process can be regarded as a link between these two processes of completely different nature.</p><p>In Paper III we construct a class of processes locally behaving as Gaussian and globally as stable processes and including the limit process obtained in Paper I. These processes can be chosen to be long-range dependent and are potentially suitable as models in applications with distinct local and global behaviour. They are defined using stochastic integrals with respect to the same compensated Poisson random measure as used in Paper II.</p>

This thesis, consisting of three papers and a summary, studies topics in the theory of stochastic processes related to long-range dependence. Much recent interest in such probabilistic models has its origin in measurements of Internet traffic data, where typical characteristics of long memory have been observed. As a macroscopic feature, long-range dependence can be mathematically studied using certain scaling limit theorems. Using such limit results, two different scaling regimes for Internet traffic models have been identified earlier. In one of these regimes traffic at large scales can be approximated by long-range dependent Gaussian or stable processes, while in the other regime the rescaled traffic fluctuates according to stable ``memoryless'' processes with independent increments. In Paper I a similar limit result is proved for a third scaling scheme, emerging as an intermediate case of the other two. The limit process here turns out to be a non-Gaussian and non-stable process with long-range dependence. In Paper II we derive a representation for the latter limit process as a stochastic integral of a deterministic function with respect to a certain compensated Poisson random measure. This representation enables us to study some further properties of the process. In particular, we prove that the process at small scales behaves like a Gaussian process with long-range dependence, while at large scales it is close to a stable process with independent increments. Hence, the process can be regarded as a link between these two processes of completely different nature. In Paper III we construct a class of processes locally behaving as Gaussian and globally as stable processes and including the limit process obtained in Paper I. These processes can be chosen to be long-range dependent and are potentially suitable as models in applications with distinct local and global behaviour. They are defined using stochastic integrals with respect to the same compensated Poisson random measure as used in Paper II.

Le sujet principal de ce mémoire est l'étude de la distribution asymptotique de la fonction f_m qui compte le nombre de diviseurs premiers distincts parmi les nombres premiers $p_1,...,p_m$. Au premier chapitre, nous présentons les sept résultats qui seront démontrés au chapitre 4. Parmi ceux-ci figurent l'analogue du théorème d'Erdos-Kac et un résultat sur les grandes déviations. Au second chapitre, nous définissons les espaces de probabilités qui serviront à calculer les probabilités asymptotiques des événements considérés, et éventuellement à calculer les densités qui leur correspondent. Le troisième chapitre est la partie centrale du mémoire. On y définit la promenade aléatoire qui, une fois normalisée, convergera vers le mouvement brownien. De là, découleront les résultats qui formeront la base des démonstrations de ceux chapitre 1. / The main topic of this masters thesis is the study of the asymptotic distribution of the fonction f_m which counts the number of distinct prime divisors among the first $m$ prime numbers, i.e. $p_1,...,p_m$. The first chapter provides the seven main results which will later on be proved in chapter 4. Among these we find the analogue of the Erdos-Kac central limit theorem and a result on large deviations. In the following chapter, we define several probability spaces on which we will calculate asymptotic probabilities of specific events. These will become necessary for calculating their corresponding densities. The third chapter is the main part of this masters thesis. In it, we introduce a random walk which, when suitably normalized, will converge to the Brownian motion. We will then obtain results which will form the basis of the proofs of those of chapiter 1.

We study combinatorial optimization problems on graphs in the mean-field model, which assigns independent and identically distributed random weights to the edges of the graph. Specifically, we focus on two generalizations of minimum weight matching on graphs. The first problem of minimum cost edge cover finds application in a computational linguistics problem of semantic projection. The second problem of minimum cost many-to-one matching appears as an intermediate optimization step in the restriction scaffold problem applied to shotgun sequencing of DNA. For the minimum cost edge cover on a complete graph on n vertices, where the edge weights are independent exponentially distributed random variables, we show that the expectation of the minimum cost converges to a constant as n →∞ For the minimum cost many-to-one matching on an n x m complete bipartite graph, scaling m as [ n/α ] for some fixed α > 1, we find the limit of the expected minimum cost as a function of α. For both problems, we show that a belief propagation algorithm converges asymptotically to the optimal solution. The belief propagation algorithm yields a near optimal solution with lesser complexity than the known best algorithms designed for optimality in worst-case settings. Our proofs use the machinery of the objective method and local weak convergence, which are ideas developed by Aldous for proving the ζ(2) limit for the minimum cost bipartite matching. We use belief propagation as a constructive proof technique to supplement the objective method. Recursive distributional equations(RDEs) arise naturally in the objective method approach. In a class of RDEs that arise as extensions of the minimum weight matching and travelling salesman problems, we prove existence and uniqueness of a fixed point distribution, and characterize its domain of attraction.

Nous présentons dans ce mémoire un nouvel algorithme de type Metropolis-Hastings dans lequel la distribution instrumentale a été conçue pour l'estimation de distributions cibles bimodales. En fait, cet algorithme peut être vu comme une modification de l'algorithme Metropolis de type marche aléatoire habituel auquel on ajoute quelques incréments de grande envergure à des moments aléatoires à travers la simulation. Le but de ces grands incréments est de quitter le mode de la distribution cible où l'on se trouve et de trouver l'autre mode. Par la suite, nous présentons puis démontrons un résultat de convergence faible qui nous assure que, lorsque la dimension de la distribution cible croît vers l'infini, la chaîne de Markov engendrée par l'algorithme converge vers un certain processus stochastique qui est continu presque partout. L'idée est similaire à ce qui a été fait par Roberts et al. (1997), mais la technique utilisée pour la démonstration des résultats est basée sur ce qui a été fait par Bédard (2006). Nous proposons enfin une stratégie pour trouver la paramétrisation optimale de notre nouvel algorithme afin de maximiser la vitesse d'exploration locale des modes d'une distribution cible donnée tout en estimant bien la pondération relative de chaque mode. Tel que dans l'approche traditionnellement utilisée pour ce genre d'analyse, notre stratégie passe par l'optimisation de la vitesse d'exploration du processus limite. Finalement, nous présentons des exemples numériques d'implémentation de l'algorithme sur certaines distributions cibles, dont une ne respecte pas les conditions du résultat théorique présenté. / In this thesis, we present a new Metropolis-Hastings algorithm whose proposal distribution has been designed to successfully estimate bimodal target distributions. This sampler may be seen as a variant of the usual random walk Metropolis sampler in which we propose large candidate steps at random times. The goal of these large candidate steps is to leave the actual mode of the target distribution in order to find the second one. We then state and prove a weak convergence result stipulating that if we let the dimension of the target distribution increase to infinity, the Markov chain yielded by the algorithm converges to a certain stochastic process that is almost everywhere continuous. The theoretical result is in the flavour of Roberts et al. (1997), while the method of proof is similar to that found in Bédard (2006). We propose a strategy for optimally parameterizing our new sampler. This strategy aims at optimizing local exploration of the target modes, while correctly estimating the relative weight of each mode. As is traditionally done in the statistical literature, our approach consists of optimizing the limiting process rather than the finite-dimensional Markov chain. Finally, we illustrate our method via numerical examples on some target distributions, one of which violates the regularity conditions of the theoretical result.