Reciprocal processes, whose concept can be traced back to E. Schrödinger, form a class of stochastic processes constructed as mixture of bridges, that satisfy a time Markov field property. We discuss here a new unifying approach to characterize several types of reciprocal processes via duality formulae on path spaces: The case of reciprocal processes with continuous paths associated to Brownian diffusions and the case of pure jump reciprocal processes associated to counting processes are treated. This presentation is based on joint works with M. Thieullen, R. Murr and C. Léonard.

We evaluated the influence of a landscape dominated by agriculture and an extensive road network on fine-scale movements of black bears (Ursus americanus) in south-central Florida. The objectives of this study were to (1) define landscape functionality including corridor use by the directionality and speed of bear movements, (2) to develop a model reflecting selected habitat characteristics during movements, (3) to identify habitat characteristics selected by bears at road-crossing locations, and (3) to develop and evaluate a predictive model for road-crossing locations based on habitat characteristics. We assessed models using GPS data from 20 adult black bears (9 F, 11 M), including 382 unique road-crossing events by 16 individuals. Directionality of bear movements were influenced by the density of cover and proximity to human infrastructure, and movement speed was influenced by density of cover and proximity to paved roads. We used the Brownian bridge movement model to assess road-crossing behavior. Landscape-level factors like density of cover and density of roads appeared more influential than roadside factors, vegetative or otherwise. Model validation procedures suggested strong predictive ability for the selected road-crossing model. These findings will allow managers to prioritize and implement sound strategies to promote connectivity and reduce road collisions.

This is an introduction to Wiener measure and the Feynman-Kac formula on general Riemannian manifolds for Riemannian geometers with little or no background in stochastics. We explain the construction of Wiener measure based on the heat kernel in full detail and we prove the Feynman-Kac formula for Schrödinger operators with bounded potentials. We also consider normal Riemannian coverings and show that projecting and lifting of paths are inverse operations which respect the Wiener measure.

Forty-four wolves in 3 boreal forest sites in Ontario were monitored via GPS radiotelemetry during 2010 and 2011 to examine spatial responses to variation in prey density. Home ranges were defined using a Brownian bridge utilization distribution, and a resource utilization function was calculated for each pack in winter and summer, based on habitat, topography, and prey density. Wolf territories were smaller where moose density was higher. Third order selection (within home range) varied by pack and season. Wolves generally selected for sloping areas, areas near water, and stands with deciduous or regenerating forest, but selected against areas with dense conifer cover. Roads were most important in summer, especially in those territories with large road networks. Habitat use in a mild winter was similar to habitat use in summer. Variable resource selection among packs emphasizes the adaptable, generalist nature of wolves even in the relatively homogenous the boreal shield. / National Science and Engineering Research Council, Ontario Graduate Scholarships, Ontario Ministry of Natural Resources - Wildlife Research and Development Section, Center for Northern Forest Ecosystem Research, Ontario Ministry of Natural Resources, Canadian Forest Service, Forest Ecosystem Science Cooperative

We apply some recent numerical solutions to diffusion bridges written in Iacus (2008). One is an approximate scheme from Bladt and S{\o}rensen (2007), another one, from  Beskos et al (2006), is an algorithm which is exact: no numerical error at given grid points!

<p>We apply some recent numerical solutions to diffusion bridges written in Iacus (2008). One is an approximate scheme from Bladt and S{\o}rensen (2007), another one, from  Beskos et al (2006), is an algorithm which is exact: no numerical error at given grid points!</p>

Ce travail de thèse porte sur la modélisation et la simulation numérique de la collision et l'agglomération de particules colloïdales dans un écoulement fluide turbulent par une nouvelle méthode. Ces particules sont sensibles dans une même mesure aux effets brownien et turbulent. La première partie du travail concerne la modélisation du phénomène physique,allant du transport des particules jusqu'à la modélisation des forces d'adhésion physico-chimiques en passant par l'étape cruciale qui est la détection des interactions entre les particules (collisions). Cette détection des collisions est dans un premier temps étudiée par rapport aux algorithmes classiques existants dans la littérature. Bien que très efficaces dans le cadre de particules soumises à l'agitation turbulente, les conclusions de cette partie exposent les limites des méthodes existantes en termes de coûts numériques, pour le traitement d'un ensemble de colloïdes soumis au mouvement brownien. La seconde partie du travail oriente alors les travaux vers une vision novatrice du phénomène physique considéré. Le caractère diffusif aléatoire est alors considéré d'un point de vu stochastique, comme un processus conditionné dans l'espace et dans le temps. Ainsi, une nouvelle méthode de détection et de traitement des collisions de particules soumises exclusivement à un mouvement diffusif est présentée et validée, exposant un gain considérable en termes de coûts numériques. Le potentiel de cette nouvelle approche est validé et ouvre de nombreuses pistes de réflexion dans l'utilisation des méthodes stochastiques appliqués à la représentation de la physique / Ph.D thesis focuses on modeling and numerical simulation of collision and agglomeration of colloidal particles in a turbulent flow by using a new method. These particles are affected by both Brownian and turbulent effects. The first part of the work deals with current models of the physical phenomenon, from the transport of single particles to a model for physico-chemical adhesive forces, and points out the critical step which is the detection of interactions between particles (collisions). This detection is initially studied by applying classical algorithms existing in the literature. Although they are very efficient in the context of particles subject to turbulent agitation, first conclusions show the limitations of these existing methods in terms of numerical costs, considering the treatment of colloids subject to the Brownian motion. The second part of this work proposes a new vision of the physical phenomenon focusing on the random diffusive behaviour. This issue is adressed from a stochastic point of view as a process conditionned in space and time. Thus, a new method for the detection and treatment of collisions is presented and validated, which represents considerable gain in terms of numerical cost. The potential of this new approach is validated and opens new opportunities for the use of stochastic methods applied to the representation of physics

The Bayesian nonparametric inference requires the construction of priors on infinite dimensional spaces such as the space of cumulative distribution functions and the space of cumulative hazard functions. Well-known priors on the space of cumulative distribution functions are the Dirichlet process, the two-parameter Poisson-Dirichlet process and the beta-Stacy process. On the other hand, the beta process is a popular prior on the space of cumulative hazard functions. This thesis is divided into three parts. In the first part, we tackle the problem of sampling from the above mentioned processes. Sampling from these processes plays a crucial role in many applications in Bayesian nonparametric inference. However, having exact samples from these processes is impossible. The existing algorithms are either slow or very complex and may be difficult to apply for many users. We derive new approximation techniques for simulating the above processes. These new approximations provide simple, yet efficient, procedures for simulating these important processes. We compare the efficiency of the new approximations to several other well-known approximations and demonstrate a significant improvement. In the second part, we develop explicit expressions for calculating the Kolmogorov, Levy and Cramer-von Mises distances between the Dirichlet process and its base measure. The derived expressions of each distance are used to select the concentration parameter of a Dirichlet process. We also propose a Bayesain goodness of fit test for simple and composite hypotheses for non-censored and censored observations. Illustrative examples and simulation results are included. Finally, we describe the relationship between the frequentist and Bayesian nonparametric statistics. We show that, when the concentration parameter is large, the two-parameter Poisson-Dirichlet process and its corresponding quantile process share many asymptotic pr operties with the frequentist empirical process and the frequentist quantile process. Some of these properties are the functional central limit theorem, the strong law of large numbers and the Glivenko-Cantelli theorem.

The Bayesian nonparametric inference requires the construction of priors on infinite dimensional spaces such as the space of cumulative distribution functions and the space of cumulative hazard functions. Well-known priors on the space of cumulative distribution functions are the Dirichlet process, the two-parameter Poisson-Dirichlet process and the beta-Stacy process. On the other hand, the beta process is a popular prior on the space of cumulative hazard functions. This thesis is divided into three parts. In the first part, we tackle the problem of sampling from the above mentioned processes. Sampling from these processes plays a crucial role in many applications in Bayesian nonparametric inference. However, having exact samples from these processes is impossible. The existing algorithms are either slow or very complex and may be difficult to apply for many users. We derive new approximation techniques for simulating the above processes. These new approximations provide simple, yet efficient, procedures for simulating these important processes. We compare the efficiency of the new approximations to several other well-known approximations and demonstrate a significant improvement. In the second part, we develop explicit expressions for calculating the Kolmogorov, Levy and Cramer-von Mises distances between the Dirichlet process and its base measure. The derived expressions of each distance are used to select the concentration parameter of a Dirichlet process. We also propose a Bayesain goodness of fit test for simple and composite hypotheses for non-censored and censored observations. Illustrative examples and simulation results are included. Finally, we describe the relationship between the frequentist and Bayesian nonparametric statistics. We show that, when the concentration parameter is large, the two-parameter Poisson-Dirichlet process and its corresponding quantile process share many asymptotic pr operties with the frequentist empirical process and the frequentist quantile process. Some of these properties are the functional central limit theorem, the strong law of large numbers and the Glivenko-Cantelli theorem.

近年來隨著法規與市場逐漸的開放，使得券商可以發行衍生性商品的種類也逐漸增加，而在眾多結構型商品中，不少商品其連結標的包含了多資產與路徑相依條款，可以看成投資一藍子股票且具有多個觀察時間的商品，一方面若連結資產上漲投資人將可得到一定的報酬，另外一方面同時具有下方保護的條款可避免本金嚴重虧損。 而此類商品包含了多資產連結且有路徑相依條款，在評價方面是一個高維度的問題，若使用傳統的蒙地卡羅法來評價，因其收斂速度緩慢常需秏費大量的計算時間，使得蒙地卡羅法在應用上有此缺點，一般來說可以使用對立變數法或控制變數法來改進收斂的速度，另外也可以使用低差異性數列即所謂的準蒙地卡羅法來改進收斂速度，並且準蒙地卡羅法與布朗橋結構或主成份分析法相結合還可加快收斂速度。 本文主要提供二種不同報酬型態的商品，第一個商品為低維度上入局商品，其報酬型態與障礙型選擇類似，第二個商品為連結多資產且路徑相依商品，以此兩商品來探討各種不同方法在不同報酬型態下的收斂速度與準確性，最後文中模擬的結果顯示在所有方法中，使用準蒙地卡羅法結合主成份分析法皆可以得到不錯的收斂速度與準確性。