The stock market has been objective of many researches that seek to identify the presence of some previsibility degree in the return series. Inside of this context grew the Market Efficiency Theory divided in three forms: weak efficiency, semi-strong and strong. The random walk hypothesis was created to test, empirically, the Market Efficiency in the weak-form. Acceptance or rejection brings implications as the possibility of its to get to foresee, somehow, based in past returns, the future returns, removing advantage of that to gain extraordinary incomes. To test the random walk, specialists in this subject they created, along the years, methods and, among these, they stand out the variance ratio tests that, initially they were applied in developed markets and, nowadays, it has also been used at emerging markets. For the development of the present research, with the intention of testing the random walk hypothesis in an emerging market (Brazil) and in a developed market (United States), were implemented the following variance ratio tests: simple, multiple, based in the ranks and signs. The returns of IBOVESPA were used, as proxy of the Brazilian stock market, and of S&P 500, to the North American market, collected daily and weekly in the period of January 03, 2000 to April 25, 2008. The results demonstrated an acceptance of the random walk hypothesis in most of the made tests appearing for a weak form of market efficiency. / O mercado de ações tem sido alvo de muitas pesquisas que visam identificar a presença de algum grau de previsibilidade nas séries de retornos. Dentro deste contexto desenvolveu-se a Teoria de Eficiência de Mercado dividida em três formas: eficiência fraca, semiforte e forte. A hipótese do caminho aleatório foi criada para testar, empiricamente, a Eficiência de Mercado na forma fraca. Sua aceitação ou rejeição traz implicâncias quanto a possibilidade de se conseguir prever, de alguma maneira, com base em retornos passados, os retornos futuros, tirando proveito disso para auferir rendimentos extraordinários. A fim de testar a hipótese do caminho aleatório estudiosos do assunto criaram, ao longo dos anos, métodos e, dentre estes, destacam-se os testes de quociente de variâncias que, inicialmente foram aplicados em mercados desenvolvidos e, atualmente, também tem sido utilizados em mercados emergentes. Para o desenvolvimento da presente pesquisa, com o intuito de testar a hipótese do caminho aleatório em um mercado emergente (Brasil) e em um mercado desenvolvido (Estados Unidos), foram aplicados os seguintes testes de quociente de variâncias: simples, múltiplas, com base nos postos e com base nos sinais. Foram utilizados os retornos do IBOVESPA, como proxy do mercado acionário brasileiro, e do S&P 500, para o mercado norte-americano, coletados diariamente e semanalmente no período de 03 de janeiro de 2000 a 25 de abril de 2008. Os resultados demonstraram uma aceitação da hipótese do caminho aleatório na maioria dos testes efetuados apontando para uma forma fraca de eficiência de mercado.

This thesis studies effective resistances of finite and infinite weighted graphs. Classical results state that it is a metric on the set of vertices of the graph and that it can be expressed completely in terms of the graph’s random walk. The first goal of this thesis is to provide a concise and accessible starting point for new scholars interested in the topic. In that spirit, we reproduce existing results and review different approaches to effective resistances using tools from several fields such as linear algebra, probability theory, geometry and functional analysis. The second goal is to characterize which metric spaces are given by the effective resistance of a graph. For the finite case, we begin by reconstructing the associated graph from the effective resistance. This leads to a complete algebraic characterization in terms of triangle inequality defects. A more geometric condition is given by showing that a metric space can only be an effective resistance if its minimal graph realization contains no incomplete cycles. We also show that our algebraic characterization can be applied to the more general theory of resistance forms as defined by Kigami. The third goal of this thesis is to investigate probabilistic representations of effective resistances. Building on the work of Tetali and Barlow, we characterize under which conditions known representations for finite graphs can be extended to infinite graphs.

Riksbanken har under senaste åren blivit kritiserade för deras bristande prognoser av svenska valutakurser. I denna uppsats undersöks det om slumpvandring (RW) är den mest framgångsrika prognosmodellen eller om alternativa ekonometriska prognosmodeller (AR, VAR och VECM) kan estimera framtida växelkurser mer korrekt på kort sikt, ett kvartal fram, och medellång sikt, fyra kvartal fram. I dessa prognosmodeller behandlas fem Svenska makroekonomiska variabler som endogena; KPI, BNP, arbetslöshet, 3 månaders statsobligationer (T-bonds), samt en exogen variabel, Amerikansk-BNP. Den data som används är kvartalsdata från första kvartalet 1993 till andra kvartalet 2020 för respektive variabel. Resultaten från studie visar på att RW är mer ackurat än de multivariata modellerna (VAR och VECM) på både kort sikt och medellång sikt. Residualerna utvärderas genom att kolla på rotmedelkvadratfel (RMSE) från respektive prognos. / In recent years, the Riksbank has been criticized for their underperforming forecasts of Swedish exchange rates. This thesis examines whether the random walk (RW) is the most successful forecasting model when forecasting the exchange rate (SEK / USD) or whether alternative economic forecasting models (AR, VAR and VECM) can estimate future exchange rates more accurately. Both in the short and medium term, one respectively four quarters ahead. In these forecast models, five Swedish macroeconomic variables are treated as endogenous; CPI, GDP, unemployment, three-month Treasury-bonds (T-Bonds), and an exogenous variable, US GDP. The data used is quarterly data from the first quarter of 1993 to the second quarter of 2020 for each variable. Results from the study show that RW is more accurate than the multivariate models (VAR and VECM) in both the short and medium term. The residuals are evaluated by looking at root mean square error (RMSE) from the respective forecast.

The efficient market hypothesis states that stock prices fully reflect availablei nformation and that stocks thereby always are priced correctly. Hence, it should be impossible to predict future prices in the stock market, and investors will gain no benefits from engaging themselves into historical analyzes. This is a quantitative study which aim to investigate if there is any difference in market efficiency in Nordic stock markets during and after the financial crisis of 2008. By applying various statistical methods, such as unitroot tests, autocorrelation tests and runs test on the returns from each country’s leading market index, the study tries to find evidence for or against the weak form of market efficiency. The study finds evidence both for and against weak form market efficiency but concludes that there is no distinct difference in market efficiency during and after the financial crisis.

The intent of this thesis is to investigate how US equity funds performance differ due to their standard deviation. In order to accomplish this study, we collected daily data for 99 US equity funds for the period 2011-2020 and divided the funds into three risk classification groups based on their standard deviation for the year 2011. The collected data was used to perform an CAPM regression and to calculate returns on a three-, five- and ten-year basis. The results for the regression and the returns for the funds was later presented as average values for the different risk classification groups. We then compared the average outcomes for the three risk classifications with each other and the index S&P 500. Our result showed that the index S&P 500 outperformed the three risk classification groups average returns for every time period. We also noticed that the difference between the average returns and the index got greater by time. We did not find any big differences between our risk classifications when it comes to their performance. Our regression analysis resulted in many negative alpha values indicating that S&P 500, as many previous studies claims, outperforms actively mutual funds. The conclusion is therefore that we could not show any evidence that the there is a major different in performance between our risk groups but also that it is difficult for fund managers to outperform index.

Dans cette thèse nous étudions un problème fondamental, le partitionnement de graphe, dans le contexte de la croissance rapide des données, le volume des données continues à augmenter, allant des réseaux sociaux à l'internet des objets. En particulier, afin de vaincre les propriétés intraitables existant dans de nombreuses graphies, par exemple, la distribution des degrés en loi de puissance, nous appliquons un nouveau mode pour coupe de sommet, à la place de la méthode traditionnelle (coupe de bord), ainsi que pour assurer une charge de travail équilibrée et raisonnablement dans le traitement de graphe distribué. En outre, pour réduire le coût de communication inter-partitions, nous proposons une méthode de partition de bord basée sur les blocs, qui peut explorer efficacement les structures graphiques sous-jacentes au niveau local. , afin d'optimiser l'exécution de l'algorithme de graphe. Par cette méthode, le temps d'exécution et des communications généraux peuvent être considérablement réduits par rapport aux approches existantes. Les challenges qui se posent dans les grands graphiques comprennent également leur grande variété. Comme nous le savons, la plupart des applications graphiques au monde réel produisent des ensembles de données hétérogènes, dans lesquels les sommets et / ou les arêtes peuvent avoir des différents types ou des différentes étiquettes. De nombreuses algorithmes de fouille de graphes sont également proposés avec beaucoup d'intérêt pour les attributs d'étiquette. Pour cette raison, notre travail est étendu aux graphes de multicouches en prenant en compte la proximité des arêtes et la distribution des étiquettes lors du processus de partitionnement. En fin de cette thèse, Nous démontré à la ses performances exceptionnelles sur les ensembles de données du monde réel. / In this thesis, we mainly focus on a fundamental problem, graph partitioning, in the context of unexpectedly fast growth of data sources, ranging from social networks to internet of things. Particularly, to conquer intractable properties existing in many graphs, e.g. power-law degree distribution, we apply the novel fashion vertex-cut, instead of the traditional edge-cut method, for achieving balanced workload in distributed graph processing. Besides, to reduce the inter-partition communication cost, we present a block-based edge partition method who can efficiently explore the locality underlying graphical structures, to enhance the execution of graph algorithm. With this method, the overhead of both communication and runtime can be decreased greatly, compared to existing approaches. The challenges arising in big graphs also include their high-variety. As we know, most of real life graph applications produce heterogenous datasets, in which the vertices and/or edges are allowed to have different types or labels. A big number of graph mining algorithms are also proposed with much concern for the label attributes. For this reason, our work is extended to multi-layer graphs with taking into account the edges closeness and labels distribution during partitioning process. Its outstanding performance over real-world datasets is demonstrated finally.

The field of mathematical modelling for pedestrian dynamics has attracted significant scientific attention, with various models proposed from perspectives such as kinetic theory, statistical mechanics, game theory and partial differential equations. Often such investigations are seen as being a part of a new branch of study in the domain of applied physics, called sociophysics. Our study proposes three models that are tailored to specific scenarios of crowd dynamics. Our research focuses on two primary issues. The first issue is centred around pedestrians navigating through a partially dark corridor that impedes visibility, requiring the calculation of the time taken for evacuation using a Markov chain model. The second issue is posed to analyse how pedestrians move through a T-shaped junction. Such a scenario is motivated by the 2022 crowd-crush disaster took place in the Itaewon district of Seoul, Korea. We propose a lattice-gas-type model that simulates pedestrians’ movement through the grid by obeying a set of rules as well as a parabolic equation with special boundary conditions. By the means of numerical simulations, we investigate a couple of evacuation scenarios by evaluating the mean velocity of pedestrians through the dark corridor, varying both the length of the obscure region and the amount of uncertainty induced by the darkness. Additionally, we propose an agent-based-modelling and cellular automata inspired model that simulates the movement of pedestrians through a T-shaped grid, varying the initial number of pedestrians. We measure the final density and time taken to reach a steady pedestrian traffic state. Finally, we propose a parabolic equation with special boundary conditions that mimic the dynamic of the pedestrian populations in a T-junction. We solve the parabolic equation using a random walk numerical scheme and compare it with a finite difference approximation. Furthermore, we prove rigorously the convergence of the random walk scheme to a corresponding finite difference scheme approximation of the solution.

Dans ce mémoire, nous nous penchons sur des résultats de localisation pour les marches biaisées en milieux aléatoires. Plus précisément, nous allons revisiter la démarche que David Croydon a fait afin de prouver un résultat de localisation pour la marche aléatoire biaisée sur la trace d’une marche aléatoire simple dans Z^d. Ensuite, nous allons débuter la généralisation de son résultat au cas où l’environnement sous-jacent est la marche aléatoire branchante par l’élaboration d’un résultat de localisation pour la marche biaisée sur la trace d’une bonne approximation des marches branchantes, leur K-squelette. / In this thesis, we will look at localization results for random walks on random environnements. More precisely, we will go through the techniques used by David Croydon to prove a localization result for the biased random walk on the trace of a simple random walk in Z^d. Then, we will begin the generalization of his result to the case where the underlying environnement is a branching random walk. To do this, we will prove a version of the localization result for the biased random walk on the trace of a K-skeleton in Z^d, which is a good approximation of the branching random walk.

In this paper we conduct an out-of-sample forecasting exercise for monthly Swedish air traveler volumes. The models considered are multiplicative seasonal ARIMA, Neural network autoregression, Exponential smoothing, the Prophet model and a Random Walk as a benchmark model. We divide the out-of-sample data into three different evaluation periods: Pre-COVID-19, during COVID-19 and Post-COVID-19 for which we calculate the MAE, MAPE and RMSE for each model in each of these evaluation periods. The results show that for the Pre-COVID-19 period all models produce accurate forecasts, in comparison to the Random Walk model. For the period during COVID-19, no model outperforms the Random Walk, with only Exponential smoothing performing as well as the Random Walk. For the period Post-COVID-19, the best performing models are Random Walk, SARIMA and Exponential smoothing, with all aforementioned models having similar performance.

L’objet de cette thèse est d’étudier plusieurs modèles probabilistes reliant les marches aléatoires et les arbres aléatoires issus de processus de branchement critiques.Dans la première partie, nous nous intéressons au modèle de marche aléatoire à valeurs dans un réseau euclidien et indexée par un arbre de Galton–Watson critique conditionné par la taille. Sous certaines hypothèses sur la loi de reproduction critique et la loi de saut centrée, nous obtenons, dans toutes les dimensions, la vitesse de croissance asymptotique du nombre de points visités par cette marche, lorsque la taille de l’arbre tend vers l’infini. Ces résultats nous permettent aussi de décrire le comportement asymptotique du nombre de points visités par une marche aléatoire branchante, quand la taille de la population initiale tend vers l’infini. Nous traitons également en parallèle certains cas où la marche aléatoire possède une dérive constante non nulle.Dans la deuxième partie, nous nous concentrons sur les propriétés fractales de la mesure harmonique des grands arbres de Galton–Watson critiques. On comprend par mesure harmonique la distribution de sortie, hors d’une boule centrée à la racine de l’arbre, d’une marche aléatoire simple sur cet arbre. Lorsque la loi de reproduction critique appartient au domaine d’attraction d’une loi stable, nous prouvons que la masse de la mesure harmonique est asymptotiquement concentrée sur une partie de la frontière, cette partie ayant une taille négligeable par rapport à celle de la frontière. En supposant que la loi de reproduction critique a une variance finie, nous arrivons à évaluer la masse de la mesure harmonique portée par un sommet de la frontière choisi uniformément au hasard. / The aim of this Ph. D. thesis is to study several probabilistic models linking the random walks and the random trees arising from critical branching processes.In the first part, we consider the model of random walk taking values in a Euclidean lattice and indexed by a critical Galton–Watson tree conditioned by the total progeny. Under some assumptions on the critical offspring distribution and the centered jump distribution, we obtain, in all dimensions, the asymptotic growth rate of the range of this random walk, when the size of the tree tends to infinity. These results also allow us to describe the asymptotic behavior of the range of a branching random walk, when the size of the initial population goes to infinity. In parallel, we treat likewise some cases where the random walk has a non-zero constant drift.In the second part, we focus on the fractal properties of the harmonic measure on large critical Galton–Watson trees. By harmonic measure, we mean the exit distribution from a ball centered at the root of the tree by simple random walk on this tree. If the critical offspring distribution is in the domain of attraction of a stable distribution, we prove that the mass of the harmonic measure is asymptotically concentrated on a boundary subset of negligible size with respect to that of the boundary. Assuming that the critical offspring distribution has a finite variance, we are able to calculate the mass of the harmonic measure carried by a random vertex uniformly chosen from the boundary.