Dans cette thèse, on étudie des processus stochastiques issus de la physique statistique. Le mouvement Brownien fractionnaire, objet central des premiers chapitres, généralise le mouvement Brownien aux cas où la mémoire est importante pour la dynamique. Ces effets de mémoire apparaissent par exemple dans les systèmes complexes et la diffusion anormale. L’absence de la propriété de Markov rend difficile l’étude probabiliste du processus. On développe une approche perturbative autour du mouvement Brownien pour obtenir de nouveaux résultats, sur des observables liées aux statistiques des extrêmes. En plus de leurs applications physiques, on explore les liens de ces résultats avec des objets mathématiques, comme les lois de Lévy et la constante de Pickands. / In this thesis, we study stochastic processes appearing in different areas of statistical physics: Firstly, fractional Brownian motion is a generalization of the well-known Brownian motion to include memory. Memory effects appear for example in complex systems and anomalous diffusion, and are difficult to treat analytically, due to the absence of the Markov property. We develop a perturbative expansion around standard Brownian motion to obtain new results for this case. We focus on observables related to extreme-value statistics, with links to mathematical objects: Levy’s arcsine laws and Pickands’ constant. Secondly, the model of elastic interfaces in disordered media is investigated. We consider the case of a Brownian random disorder force. We study avalanches, i.e. the response of the system to a kick, for which several distributions of observables are calculated analytically. To do so, the initial stochastic equation is solved using a deterministic non-linear instanton equation. Avalanche observables are characterized by power-law distributions at small-scale with universal exponents, for which we give new results.

<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.

<p>This study examines the behaviour of the South African financial markets with regards to the Geometric Brownian motion process. It uses the daily, weekly, and monthly stock returns time series of some major securities trading in the South African financial market, more specifically the US dollar/Euro, JSE ALSI Total Returns Index, South African All Bond Index, Anglo American Corporation, Standard Bank, Sasol, US dollar Gold Price , Brent spot oil price, and South African white maize near future. The assumptions underlying the&nbsp / Geometric Brownian motion in finance, namely the stationarity, the normality and the independence of stock returns, are tested using both graphical (histograms and normal plots)&nbsp / and statistical test (Kolmogorov-Simirnov test, Box-Ljung statistic and Augmented Dickey-Fuller test) methods to check whether or not the Brownian motion as a model for South&nbsp / African financial markets holds. The Hurst exponent or independence index is also applied to support the results from the previous test. Theoretically, the independent or Geometric&nbsp / Brownian motion time series should be characterised by the Hurst exponent of &frac12 / . A value of a Hurst exponent different from that would indicate the presence of long memory or&nbsp / fractional Brownian motion in a time series. The study shows that at least one assumption is violated when the Geometric Brownian motion process is examined assumption by&nbsp / assumption. It also reveals the presence of both long memory and random walk or Geometric Brownian motion in the South African financial markets returns when the Hurst index analysis is used and finds that the Currency market is the most efficient of the South African financial markets. The study concludes that although some assumptions underlying the&nbsp / rocess are violated, the Brownian motion as a model in South African financial markets can not be rejected. It can be accepted in some instances if some parameters such as the Hurst exponent are added.</p>

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.

<p>This study examines the behaviour of the South African financial markets with regards to the Geometric Brownian motion process. It uses the daily, weekly, and monthly stock returns time series of some major securities trading in the South African financial market, more specifically the US dollar/Euro, JSE ALSI Total Returns Index, South African All Bond Index, Anglo American Corporation, Standard Bank, Sasol, US dollar Gold Price , Brent spot oil price, and South African white maize near future. The assumptions underlying the&nbsp / Geometric Brownian motion in finance, namely the stationarity, the normality and the independence of stock returns, are tested using both graphical (histograms and normal plots)&nbsp / and statistical test (Kolmogorov-Simirnov test, Box-Ljung statistic and Augmented Dickey-Fuller test) methods to check whether or not the Brownian motion as a model for South&nbsp / African financial markets holds. The Hurst exponent or independence index is also applied to support the results from the previous test. Theoretically, the independent or Geometric&nbsp / Brownian motion time series should be characterised by the Hurst exponent of &frac12 / . A value of a Hurst exponent different from that would indicate the presence of long memory or&nbsp / fractional Brownian motion in a time series. The study shows that at least one assumption is violated when the Geometric Brownian motion process is examined assumption by&nbsp / assumption. It also reveals the presence of both long memory and random walk or Geometric Brownian motion in the South African financial markets returns when the Hurst index analysis is used and finds that the Currency market is the most efficient of the South African financial markets. The study concludes that although some assumptions underlying the&nbsp / rocess are violated, the Brownian motion as a model in South African financial markets can not be rejected. It can be accepted in some instances if some parameters such as the Hurst exponent are added.</p>

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.

Vilela Mendes et al. (2015), based on the discovery of long-range dependence in the volatility of stock returns, proposed a stochastic volatility continuous mar- ket model where the volatility is given as a transform of the fractional Brownian motion (fBm) and studied its No-Arbitrage and completeness properties under va- rious assumptions. We investigate the possibility of generalization of their results from fBm to a wider class of Hermite processes. We have reworked and completed the proofs of the propositions in the cited article. Under the assumption of indepen- dence of the stock price and volatility driving processes the model is arbitrage-free. However, apart from a case of a special relation between the drift and the volatility, the model is proved to be incomplete. Under a different assumption that there is only one source of randomness in the model and the volatility driving process is bounded, the model is arbitrage-free and complete. All the above results apply to any Hermite process driving the volatility. 1

Magister Scientiae - MSc / This study examines the behaviour of the South African financial markets with regards to the Geometric Brownian motion process. It uses the daily, weekly, and monthly stock returns time series of some major securities trading in the South African financial market, more specifically the US dollar/Euro, JSE ALSI Total Returns Index, South African All Bond Index, Anglo American Corporation, Standard Bank, Sasol, US dollar Gold Price , Brent spot oil price, and South African white maize near future. The assumptions underlying the Geometric Brownian motion in finance, namely the stationarity, the normality and the independence of stock returns, are tested using both graphical (histograms and normal plots) and statistical test (Kolmogorov-Simirnov test, Box-Ljung statistic and Augmented Dickey-Fuller test) methods to check whether or not the Brownian motion as a model for South African financial markets holds. The Hurst exponent or independence index is also applied to support the results from the previous test. Theoretically, the independent or Geometric Brownian motion time series should be characterised by the Hurst exponent of ½. A value of a Hurst exponent different from that would indicate the presence of long memory or fractional Brownian motion in a time series. The study shows that at least one assumption is violated when the Geometric Brownian motion process is examined assumption by assumption. It also reveals the presence of both long memory and random walk or Geometric Brownian motion in the South African financial markets returns when the Hurst index analysis is used and finds that the Currency market is the most efficient of the South African financial markets. The study concludes that although some assumptions underlying the rocess are violated, the Brownian motion as a model in South African financial markets can not be rejected. It can be accepted in some instances if some parameters such as the Hurst exponent are added. / South Africa

Cette thèse porte sur l’étude de la mémoire longue de la volatilité des rendements d’actions. Dans une première partie, nous apportons une interprétation de la mémoire longue en termes de comportement d’agents grâce à un modèle de volatilité à mémoire longue dont les paramètres sont reliés aux comportements hétérogènes des agents pouvant être rationnels ou à rationalité limitée. Nous déterminons de manière théorique les conditions nécessaires à l’obtention de mémoire longue. Puis nous calibrons notre modèle à partir des séries de volatilité réalisée journalière d’actions américaines de moyennes et grandes capitalisations et observons le changement de comportement des agents entre la période précédant l’éclatement de la bulle internet et celle qui la suit. La deuxième partie est consacrée à la prise en compte de la mémoire longue en gestion de portefeuille. Nous commençons par proposer un modèle de choix de portefeuille à volatilité stochastique dans lequel la dynamique de la log-volatilité est caractérisée par un processus d’Ornstein-Uhlenbeck. Nous montrons que l’augmentation du niveau d’incertitude sur la volatilité future induit une révision du plan de consommation et d’investissement. Puis dans un deuxième modèle, nous introduisons la mémoire longue grâce au mouvement brownien fractionnaire. Cela a pour conséquence de transposer le système économique d’un cadre markovien à un cadre non-markovien. Nous fournissons donc une nouvelle méthode de résolution fondée sur la technique de Monte Carlo. Puis, nous montrons toute l’importance de modéliser correctement la volatilité et mettons en garde le gérant de portefeuille contre les erreurs de spécification de modèle. / This PhD thesis is about the study of the long memory of the volatility of asset returns. In a first part, we bring an interpretation of long memory in terms of agents’ behavior through a long memory volatility model whose parameters are linked with the bounded rational agents’ heterogeneous behavior. We determine theoretically the necessary condition to get long memory. Then we calibrate our model from the daily realized volatility series of middle and large American capitalization stocks. Eventually, we observe the change in the agents’ behavior between the period before the internet bubble burst and the one after. The second part is devoted to the consideration of long memory in portfolio management. We start by suggesting a stochastic volatility portfolio model in which the dynamics of the log-volatility is characterized by an Ornstein-Uhlenbeck process. We show that when the uncertainty of the future volatility level increases, it induces the revision of the consumption and investment plan. Then in a second model, we introduce a long memory component by the use of a fractional Brownian motion. As a consequence, it transposes the economic system from a Markovian framework to a non-Markovian one. So we provide a new resolution method based on Monte Carlo technique. Then we show the high importance to well model the volatility and warn the portfolio manager against the misspecification errors of the model.