This paper explores different covariance matrix estimators in application to geometric Brownian motion. Particular interest is given to shrinkage estimation methods. In collaboration with Söderberg & Partners risk management team, the goal is to find an estimation that performs well in low-data scenarios and is robust against erroneous model assumptions, particularly the Gaussian assumption of the stock price distribution. Estimations are compared by two criteria: Frobenius norm distance between the estimate and the true covariance matrix, and the condition number of the estimate. By considering four estimates — the sample covariance matrix, Ledoit-Wolf, Tyler M-estimator, and a novel Tyler-Ledoit-Wolf (TLW) estimator — this paper concludes that the TLW estimator performs best when considering the two criteria.

Stochastic delay differential equations (SDDEs) are systems of differential equations with a time lag in a noisy or random environment. There are many nonlinear SDDEs where a linear growth condition is not satisfied, for example, the stochastic delay Lotka-Volterra model of food chain discussed by Xuerong Mao and Martina John Rassias in 2005. Much research has been done using discrete delay where the dynamics of a process at time t depend on the state of the process in the past after a single fixed time lag \tau. We are researching processes with continuously distributed delay which depend on weighted averages of past states over the entire time lag interval [t-\tau,t].By using martingale concepts, we prove sufficient conditions for the existence of a unique solution, ultimate boundedness, and non-extinction of one-dimensional nonlinear SDDE with continuously distributed delay. We give generalized Khasminskii-type conditions which along with local Lipschitz conditions are sufficient to guarantee the existence of a unique global solution of certain n-dimensional nonlinear SDDEs with continuously distributed delay. Further, we give conditions under which Euler-Maruyama numerical approximations of such nonlinear SDDEs converge in probability to their exact solutions.We give some examples of one-dimensional and 2-dimensional stochastic differential equations with continuously distributed delay which satisfy the sufficient conditions of our theorems. Moreover, we simulate their solutions and analyze the error of approximation using MATLAB to implement the Euler-Maruyama algorithm.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第25093号 / 理博第5000号 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 Croydon David Alexander, 教授 大木谷 耕司, 准教授 梶野 直孝 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

After a review of the notions of Hausdorff and Fourier dimensions from fractal geometry and Fourier analysis and the properties of local times of Brownian motion, we study the Fourier structure of Brownian level sets. We show that if δa(X) is the Dirac measure of one-dimensional Brownian motion X at the level a, that is the measure defined by the Brownian local time La at level a, and μ is its restriction to the random interval [0, L−1 a (1)], then the Fourier transform of μ is such that, with positive probability, for all 0 ≤ β < 1/2, the function u → |u|β|μ(u)|2, (u ∈ R), is bounded. This growth rate is the best possible. Consequently, each Brownian level set, reduced to a compact interval, is with positive probability, a Salem set of dimension 1/2. We also show that the zero set of X reduced to the interval [0, L−1 0 (1)] is, almost surely, a Salem set. Finally, we show that the restriction μ of δ0(X) to the deterministic interval [0, 1] is such that its Fourier transform satisfies E (|ˆμ(u)|2) ≤ C|u|−1/2, u 6= 0 and C > 0. Key words: Hausdorff dimension, Fourier dimension, Salem sets, Brownian motion, local times, level sets, Fourier transform, inverse local times. / Decision Sciences / PhD. (Operations Research)

Contexte : Gesneria et Rhytidophyllum (Gesneriaceae) sont deux genres de plantes Antillais aillant subi une forte diversification et qui présentent une forte variabilité de modes de pollinisation associés à des traits floraux particuliers. Les spécialistes des colibris ont des fleurs tubulaires rouges, alors que les spécialistes des chauves-souris et les généralistes présentent des fleurs campanulées de couleur pâle. La capacité d’être pollinisé par des chauves-souris (en excluant les colibris ou en devenant généraliste) a évolué plusieurs fois indépendamment au sein du groupe. Ces caractéristiques font de ces plantes un bon modèle pour étudier les relations entre l’évolution des modes de pollinisation et la diversification spécifique et écologique. Pour ceci, nous avons étudié les bases génétiques des changements de mode de pollinisation et les liens entre ces modes de pollinisations et la diversification des niches bioclimatiques. Méthodes : Nous avons réalisé une étude de QTLs pour caractériser les régions génomiques associées à la transition de syndrome de pollinisation entre une espèce à stratégie de pollinisation mixte (Rhytidophyllum auriculatum) et une espèce spécialiste des colibris (Rhytidophyllum rupincola). Nous avons parallèlement analysé les relations entre les changements de modes de pollinisation (dimension biotique de la niche écologique) et l’évolution des niches bioclimatiques chez ces plantes. Enfin, d’un point de vue théorique, nous avons testé l’effet de la fréquence et de l’amplitude des changements environnementaux sur les patrons d’évolution des niches écologiques. Résultats : L’étude des QTLs a montré que la couleur et le volume de nectar sont basés chacun sur un QTL majeur, alors que la forme de la corolle a une base génétique plus complexe. Par ailleurs ces différents QTLs ne sont pas liés physiquement dans le génome. L’analyse des niches bioclimatiques a montré que ces Gesneriaceae antillaises sont caractérisées par un conservatisme phylogénétique de niche bioclimatique (PNC) et que l’évolution de ces niches est indépendante des stratégies de pollinisation. Les plantes semblent aussi être relativement généralistes du point de vue de leur niche abiotique. Finalement, nous avons testé l’hypothèse selon laquelle l’adaptation à un environnement temporellement hétérogène pourrait expliquer à la fois le caractère généraliste des plantes et leur patron de PNC. Cette hypothèse s’est trouvée partiellement vérifiée. Conclusion : Si l’indépendance génétique des traits floraux a pu faciliter l’émergence des syndromes de pollinisation en réduisant les contraintes génétiques, il semble que la répartition largement chevauchante des colibris et des chauves-souris ne représente pas une opportunité écologique suffisante pour expliquer les évolutions répétées vers la pollinisation par les chauves-souris. En revanche, les perturbations environnementales causant régulièrement des déclins dans les populations de pollinisateurs pourraient expliquer l’avantage des plantes qui ont une stratégie de pollinisation mixte. / Background: Gesneria and Rhytidophyllum (Gesneriaceae) are two genera endemic to the Antilles that underwent an important diversification and that present a great vari- ability in pollination modes with regard to specific floral traits. Hummingbird specialists harbour red tubular flowers while bat specialists and generalists have campanulate (i.e., bell shaped) flowers with pale colours. Bat pollination (excluding or not hummingbirds) evolved multiple times independently in this group. These plants are thus a good model to study the relationship between the evolution of pollination mode and ecological and species diversification. To understand these relationships, we studied the genetic basis of pollination mode transition and the link between pollination mode and bioclimatic niches diversification. Methods: We performed a QTL analysis to detect genomic regions underlying the floral traits involved in the pollination syndrome transition between Rhytidophyllum auriculatum (a generalist species) and Rhytidophyllum rupincola (a hummingbird specialist). Also, we analysed the consequence of pollination mode transitions (which represent the biotic part of ecological niches) on bioclimatic niches evolution in Gesneria and Rhytidophyllum. Then, we tested whether environmental changes can result in patterns of phylogenetic bioclimatic niche conservatism through time. Results: The QTLs analysis showed that corolla colour and nectar volume are both based on one major QTL, while corolla shape is determined by a more complex genetic architecture involving several unlinked QTLs. These Antillean Gesneriaceae were found to have a pattern of phylogenetic (bioclimatic) niche conservatism (PNC) and their niche evolution was found to be independent from pollination strategies. Overall, the plants were found to have relatively widespread bioclimatic niches. Finally, we partially confirmed the hypothesis that adapting to temporally variable environment might cause both species generalization and PNC pattern. Conclusion: Genetic independence of floral traits might have facilitated pollination syn- dromes evolution by reducing genetic constraints. However, the overlapping distribution of hummingbirds and bats do not represent an ecological opportunity that could explain re- peated evolutions toward bat pollination. However, environmental perturbations causing regular pollinator populations collapses could explain the advantage for plants to favour generalist strategies.

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 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>

<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>

After a review of the notions of Hausdorff and Fourier dimensions from fractal geometry and Fourier analysis and the properties of local times of Brownian motion, we study the Fourier structure of Brownian level sets. We show that if δa(X) is the Dirac measure of one-dimensional Brownian motion X at the level a, that is the measure defined by the Brownian local time La at level a, and μ is its restriction to the random interval [0, L−1 a (1)], then the Fourier transform of μ is such that, with positive probability, for all 0 ≤ β < 1/2, the function u → |u|β|μ(u)|2, (u ∈ R), is bounded. This growth rate is the best possible. Consequently, each Brownian level set, reduced to a compact interval, is with positive probability, a Salem set of dimension 1/2. We also show that the zero set of X reduced to the interval [0, L−1 0 (1)] is, almost surely, a Salem set. Finally, we show that the restriction μ of δ0(X) to the deterministic interval [0, 1] is such that its Fourier transform satisfies E (|ˆμ(u)|2) ≤ C|u|−1/2, u 6= 0 and C > 0. Key words: Hausdorff dimension, Fourier dimension, Salem sets, Brownian motion, local times, level sets, Fourier transform, inverse local times. / Decision Sciences / PhD. (Operations Research)

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