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Automates cellulaires probabilistes et processus itérés ad libitum / Probabilistic cellular automata and processes iterated ad libitumCasse, Jérôme 19 November 2015 (has links)
La première partie de cette thèse porte sur les automates cellulaires probabilistes (ACP) sur la ligne et à deux voisins. Pour un ACP donné, nous cherchons l'ensemble de ces lois invariantes. Pour des raisons expliquées en détail dans la thèse, ceci est à l'heure actuelle inenvisageable de toutes les obtenir et nous nous concentrons, dans cette thèse, surles lois invariantes markoviennes. Nous établissons, tout d'abord, un théorème de nature algébrique qui donne des conditions nécessaires et suffisantes pour qu'un ACP admette une ou plusieurs lois invariantes markoviennes dans le cas où l'alphabet E est fini. Par la suite, nous généralisons ce résultat au cas d'un alphabet E polonais après avoir clarifié les difficultés topologiques rencontrées. Enfin, nous calculons la fonction de corrélation du modèleà 8 sommets pour certaines valeurs des paramètres du modèle en utilisant une partie desrésultats précédents. / The first part of this thesis is about probabilistic cellular automata (PCA) on the line and with two neighbors. For a given PCA, we look for the set of its invariant distributions. Due to reasons explained in detail in this thesis, it is nowadays unthinkable to get all of them and we concentrate our reections on the invariant Markovian distributions. We establish, first, an algebraic theorem that gives a necessary and sufficient condition for a PCA to have one or more invariant Markovian distributions when the alphabet E is finite. Then, we generalize this result to the case of a polish alphabet E once we have clarified the encountered topological difficulties. Finally, we calculate the 8-vertex model's correlation function for some parameters values using previous results.The second part of this thesis is about infinite iterations of stochastic processes. We establish the convergence of the finite dimensional distributions of the α-stable processes iterated n times, when n goes to infinite, according to parameter of stability and to drift r. Then, we describe the limit distributions. In the iterated Brownian motion case, we show that the limit distributions are linked with iterated functions system.
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Asymptotique des solutions d'équations différentielles de type frottement perturbées par des bruits de Lévy stables / Asymptotic of solutions of friction type differential equations disturbed by stable Lévy noiseÉon, Richard 05 July 2016 (has links)
Cette thèse porte sur l'étude d'équations différentielles de type frottement, c'est à dire d'équations de type attractive, avec un unique point stable 0, caractérisant la vitesse d'un objet soumis à une force de frottement. La vitesse de cet objet subit des perturbations aléatoires de type Lévy. Dans une première partie, nous nous intéressons aux propriétés fondamentales de ces EDS : existence et unicité de la solution, caractère markovien et ergodique de celle-ci et plus particulièrement le cas des processus de Lévy stable. Dans une deuxième partie, nous étudions la stabilité de la solution de ces EDS lorsque la perturbation est un processus de Lévy stable qui tend vers 0. En effet, nous démontrons l'existence d'un développement limité d'ordre un autour de la solution déterministe pour la vitesse et la position de l'objet. Dans une troisième partie, nous étudions le comportement asymptotique des solutions lorsque la vitesse initiale est nulle et que la perturbation est un processus de Lévy stable symétrique. Nous prouvons dans cette partie que l'accumulation de perturbations entraîne un comportement asymptotique gaussien de la position de l'objet, à condition que l'indice de stabilité du processus de Lévy et la croissance du potentiel soient suffisamment grand. Dans une quatrième partie, nous levons l'hypothèse de symétrie de la perturbation en démontrant le même résultat que dans la troisième partie mais avec une dérive. Pour cela, nous étudions tout d'abord la queue de distribution de la mesure invariante associée à la vitesse de l'objet. Enfin dans une dernière partie, nous nous intéressons au résultat de la troisième partie lorsque la perturbation est la somme d'un mouvement brownien et d'un processus de Lévy purement à sauts. Puis nous commençons l'étude de la dimension deux en traitant le cas où les équations sont découplées mais où les mouvement brownien directeurs sont dépendants. / This thesis deals with the study of friction type differential equations, in other words, attractive equations, with a unique stable point 0, describing the speed of an object submitted to a frictional force. This object's speed is disturbed by Lévy type random perturbations. In a first part, one is interested in fondamental properties of these SDE: existence and unicity of a solution, Markov and ergodic properties, and more particularly the case of stable Lévy processes.In a second part, one study the stability of the solution of these SDE when the perturbation is an stable Lévy process that tends to 0. In fact, one proves the existence of a Taylor expansion of order one around the deterministic solution for the object's speed and position. In a third part, one study the asymptotic behaviour of the solutions when the initial speed is 0 and the perturbation is a symmetric stable Lévy process. One proves that the amount of perturbations, if the stability's index of the Lévy process and the increasing of the potential are big enough, leads to a gaussian asymptotic behaviour for the object's position.In a forth part, one relaxes the assumption of symmetry of the perturbation by proving the same result as in the third part but with a drift. To do so, one first studies the tail of the invariant measure of the object's speed.Finally, in a last part, one is interested in the same result as in the third part when the perturbation is the sum of the Brownian motion and a pure jump stable Lévy process. Then, one begins the study of the dimension two by considering the case where the equations are separated but where the driving Brownian motions are dependent.
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A Non-Gaussian Limit Process with Long-Range DependenceGaigalas, Raimundas January 2004 (has links)
<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>
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A Non-Gaussian Limit Process with Long-Range DependenceGaigalas, Raimundas January 2004 (has links)
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.
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Multifractal Analysis for the Stock Index Futures Returns with Wavelet Transform Modulus Maxima / 股價指數期貨報酬率的多重碎形分析與小波轉換的模數最大值洪榕壕, Hung,Jung-Hao Unknown Date (has links)
本文應用資產報酬率的多重碎形模型,該模型為一整合財務時間序列上的厚尾及波動持續性的連續時間過程。多重碎形的方法允許我們估計隨時間變動的報酬率高階動差,進而推論財務時間序列的產生機制。我們利用小波轉換的模數最大值計算多重碎形譜,透過譜分解得到資產報率分配的高階動差資訊。根據實證結果,我們得到S&P和DJIA的股價指數期貨報酬率符合動差尺度行為且資料也展現幕律的形態。根據估計出的譜形態為對數常態分配。實證結果也顯示S&P和DJIA的股價指數期貨報酬率均具有長記憶及多重碎形的特性。 / We apply the multifractal model of asset returns (MMAR), a class of continuous-time processes that incorporate the thick tails and volatility persistence of financial time series. The multifractal approach allows for higher moments of returns that may vary with the time horizon and leads to infer about the generating mechanism of the financial time series. The multifractal spectrum is calculated by the Wavelet Transform Modulus Maxima (WTMM) provides information on the higher moments of the distribution of asset returns and the multiplicative cascade of volatilities. We obtain the evidences of multifractality in the moment-scaling behavior of S&P and DJIA stock index futures returns and the moments of the data represent a power law. According to the shape of the estimated spectrum we infer a log normal distribution.The empirical evidences show that both of them have long memory and multifractal property.
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會計揭露對於市場風險之資訊內涵 / How informative are accounting disclosures about market risk?魏向璟, Wei,Hsiang-Chin Unknown Date (has links)
基於SEC之要求,越來越多美國金融機構於其財務報表附註中揭露金融交易資產之風險值;然而計算風險值涉及到許多假設,於是導致過去部分文獻對於風險值資訊揭露之可靠性產生質疑。本研究以風險值之揭露對於報表使用者之資訊價值作為研究課題;為求與附註揭露之風險值資訊比較,本研究以公司帳列之金融交易資產(Trading Assets)、金融交易收入(Trading Revenue)為基礎,利用蒙地卡羅模擬法模擬帳列金融交易資產於次期可能產生之最大潛在損失,並且透過OLS regression及panel data model探討:
(1)風險值及金融交易資產潛在可能損失是否可以預測次期金融交易收入波動
(2)風險值與金融交易資產潛在可能損失資訊之提供是否影響次期股票交易量
(3)風險值與金融交易資產潛在可能損失資訊是否可以有效預測次期股價報酬 率變異。
實證結果顯示,風險值之揭露與金融交易資產潛在可能損失之資訊對於預計次期金融交易收入之波動與股價報酬率變異均呈現顯著正相關;易言之,上述兩種資訊之揭露均提供增額之資訊價值。惟另方面,風險值之揭露與金融交易資產潛在可能損失之資訊卻與次期股票交易量呈正相關,也就是說上述兩種資訊的揭露反而會造成投資人降低長期投資持有的意願。 / Financial institutions in the United States are required by the Securities and Exchange Commission to disclose their Value at Risk (VaR) in the footnotes of the financial statements. Over the years, VaR has been used by institutional investors, industry consultants, and regulators as one of the key measures of market risk. However, there are a number of approaches to calculating VaR and some of them may involve various statistical models and assumptions. Due to the fact that different models and assumptions may be used, the VaR numbers produced by different institutions are difficult to compare with one another. The usefulness of these numbers is therefore decreased.
This thesis examines the usefulness of disclosures of VaR information. In order to compare with VaR disclosures, the implied potential maximum losses of trading assets are simulated by using Monte Carlo simulation. We then employ the OLS regression and the panel data models to investigate the following research questions:
(1)whether VaR and implied potential maximum losses of trading assets disclosed by financial institutions are instrumental in predicting the variability of trading revenue for the next quarter;
(2)whether VaR and implied potential maximum losses of trading assets disclosed by financial institutions affect investors' investing decision;
(3)how useful are VaR and implied potential maximum losses of trading assets in predicting the volatility of daily stock return next quarter.
The empirical results indicate that VaR and implied potential maximum losses of trading assets are significantly related to the variability of trading revenue and the volatility of daily stock returns next quarter. This evidence suggests that both types of disclosures provide incremental information on predicting the variability of trading revenue and investment risk in the future. Nevertheless, we also find that both VaR disclosures and implied potential maximum losses of trading assets are positively associated with future average stock trading volume, implying that investors tend to trade stock more frequently when the market risk information is disclosed.
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Nonequilibrium fluctuations of a Brownian particleGomez-Solano, Juan Rubén 08 November 2011 (has links) (PDF)
This thesis describes an experimental study on fluctuations of a Brownian particle immersed in a fluid, confined by optical tweezers and subject to two different kinds of non-equilibrium conditions. We aim to gain a rather general understanding of the relation between spontaneous fluctuations, linear response and total entropy production for processes away from thermal equilibrium. The first part addresses the motion of a colloidal particle driven into a periodic non-equilibrium steady state by a nonconservative force and its response to an external perturbation. The dynamics of the system is analyzed in the context of several generalized fluctuation-dissipation relations derived from different theoretical approaches. We show that, when taking into account the role of currents due to the broken detailed balance, the theoretical relations are verified by the experimental data. The second part deals with fluctuations and response of a Brownian particle in two different aging baths relaxing towards thermal equilibrium: a Laponite colloidal glass and an aqueous gelatin solution. The experimental results show that heat fluxes from the particle to the bath during the relaxation process play the same role of steady state currents as a non-equilibrium correction of the fluctuation-dissipation theorem. Then, the present thesis provides evidence that the total entropy production constitutes a unifying concept which links the statistical properties of fluctuations and the linear response function for non-equilibrium systems either in stationary or non stationary states.
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Distribution asymptotique du nombre de diviseurs premiers distincts inférieurs ou égaux à mPersechino, Roberto 05 1900 (has links)
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.
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Stochastic Modelling of Financial Processes with Memory and Semi-Heavy TailsPesee, Chatchai January 2005 (has links)
This PhD thesis aims to study financial processes which have semi-heavy-tailed marginal distributions and may exhibit memory. The traditional Black-Scholes model is expanded to incorporate memory via an integral operator, resulting in a class of market models which still preserve the completeness and arbitragefree conditions needed for replication of contingent claims. This approach is used to estimate the implied volatility of the resulting model. The first part of the thesis investigates the semi-heavy-tailed behaviour of financial processes. We treat these processes as continuous-time random walks characterised by a transition probability density governed by a fractional Riesz- Bessel equation. This equation extends the Feller fractional heat equation which generates a-stable processes. These latter processes have heavy tails, while those processes generated by the fractional Riesz-Bessel equation have semi-heavy tails, which are more suitable to model financial data. We propose a quasi-likelihood method to estimate the parameters of the fractional Riesz- Bessel equation based on the empirical characteristic function. The second part considers a dynamic model of complete financial markets in which the prices of European calls and puts are given by the Black-Scholes formula. The model has memory and can distinguish between historical volatility and implied volatility. A new method is then provided to estimate the implied volatility from the model. The third part of the thesis considers the problem of classification of financial markets using high-frequency data. The classification is based on the measure representation of high-frequency data, which is then modelled as a recurrent iterated function system. The new methodology developed is applied to some stock prices, stock indices, foreign exchange rates and other financial time series of some major markets. In particular, the models and techniques are used to analyse the SET index, the SET50 index and the MAI index of the Stock Exchange of Thailand.
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Vybrané geometrické vlastnosti trajektorií Brownova pohybu / On Selected Geometric Properties of Brownian Motion PathsHonzl, Ondřej January 2012 (has links)
Title: On Selected Geometric Properties of Brownian Motion Paths Author: Mgr. Ondřej Honzl E-mail Address: honzl@karlin.mff.cuni.cz Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Jan Rataj, CSc. E-mail Address: rataj@karlin.mff.cuni.cz Department: Mathematical Institute, Charles University Abstract: Our thesis is focused on certain geometric properties of Brownian motion paths. Firstly, it deals with cone points of Brownian motion in the plane and we show some connections between cone points and critical points of Brownian motion. The motivation of the study of critical points is provided by a pleasant behavior of the distance function outside of the set of these points. We prove the theorem on a non-existence of π+ cone points on fixed line. This statement leads us to the conjecture that there are only countably many critical points of the Brownian motion path in the plane. Next, the thesis discusses an asymptotic behavior of the surface area of r-neigh- bourhood of Brownian motion, which is called Wiener sausage. Using the proper- ties of a Kneser function, we prove the claim about the relation of the Minkowski content and S-content. As the consequence, we obtain a limit behavior of the surface area of the Wiener sausage almost surely in dimension d ≥ 3. Finally,...
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