We quantify the performance of approximations to stochastic filtering by the Kullback- Leibler divergence to the optimal Bayesian filter. Using a two-state Markov process that drives a Brownian measurement process as prototypical test case, we compare two stochastic filtering approximations: a static low-pass filter as baseline, and machine learning of Volterra expansions using nonlinear Vector Auto-Regression (nVAR). We highlight the crucial role of the chosen performance metric, and present two solutions to the specific challenge of predicting a likelihood bounded between 0 and 1.

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Several manufacturers of road vehicles today are working on developing autonomous vehicles. One subject that is often up for discussion when it comes to integrating autonomous road vehicles into the infrastructure is the safety aspect. There is in the context no common view of how safety should be quantified. As a contribution to this discussion we propose describing each potential hazardous event of a vehicle as a Semi-Markov Process (SMP). A reliability-based method for using the semi-Markov representation to calculate the probability of a hazardous event to occur is presented. The method simplifies the expression for the reliability using the Laplace-Stieltjes transform and calculates the transform of the reliability exactly. Numerical inversion algorithms are then applied to approximate the reliability up to a desired error tolerance. The method is validated using alternative techniques and is thereafter applied to a system for automated steering based on a real example from the industry. A desired evolution of the method is to involve a framework for how to represent each hazardous event as a SMP. / Flertalet tillverkare av vägfordon jobbar idag på att utveckla autonoma fordon. Ett ämne ofta på agendan i diskussionen om att integrera autonoma fordon på vägarna är säkerhet. Det finns i sammanhanget ingen klar bild över hur säkerhet ska kvantifieras. Som ett bidrag till denna diskussion föreslås här att beskriva varje potentiellt farlig situation av ett fordon som en Semi-Markov process (SMP). En metod presenteras för att via beräkning av funktionssäkerheten nyttja semi-Markov representationen för att beräkna sannolikheten för att en farlig situation ska uppstå. Metoden nyttjar Laplace-Stieltjes transformen för att förenkla uttrycket för funktionssäkerheten och beräknar transformen av funktionssäkerheten exakt. Numeriska algoritmer för den inversa transformen appliceras sedan för att beräkna funktionssäkerheten upp till en viss feltolerans. Metoden valideras genom alternativa tekniker och appliceras sedan på ett system för autonom styrning baserat på ett riktigt exempel från industrin. En fördelaktig utveckling av metoden som presenteras här skulle vara att involvera ett ramverk för hur varje potentiellt farlig situation ska representeras som en SMP.

本論文中，我們提出基於機場安全檢查的分層排隊理論模型，模型中的旅客基於歷史的安全數據被分成三組。我們運用二維馬可夫過程（two-dimensional Markov process）以及馬可夫調控卜瓦松過程（Markov modulated Poisson process）構建模型的排隊系統並加以分析。我們收集了台灣桃園國際機場和其它兩個機場的旅客數據以驗證我們提出的模型，並運用模擬退火法（simulated annealing）求得近似最佳解（near-optimum solution）。最後我們通過模型的旅客平均等候時間和另外兩種等候模型進行比較，之後得出我們的模型確實可以在不增加成本，甚至提升安全性的同時能夠有效地減少平均等候時間。 / This thesis proposes a tiered inspection system for airport security, wherein passengers are divided into three classes based on historical security records. A two-dimensional Markov process and a Markov modulated Poisson process (MMPP) queue were used in the formulation of the security inspection system. Simulated annealing was then used to obtain near-optimum solution for the model. The efficacy of the proposed model was evaluated using the arrival data of passengers at Taoyuan International Airport and other two international airports. A comparison with two conventional queueing models with regard to the average waiting time demonstrated the effectiveness of the proposed security inspection system in enhancing service efficiency and boosting the level of security.

As ruin theory evolves in recent years, there has been a variety of quantities pertaining to an insurer's bankruptcy at the centre of focus in the literature. Despite the fact that these quantities are distinct from each other, it was brought to our attention that many solution methods apply to nearly all ruin-related quantities. Such a peculiar similarity among their solution methods inspired us to search for a general form that reconciles those seemingly different ruin-related quantities. The stochastic approach proposed in the thesis addresses such issues and contributes to the current literature in three major directions. (1) It provides a new function that unifies many existing ruin-related quantities and that produces more new quantities of potential use in both practice and academia. (2) It applies generally to a vast majority of risk processes and permits the consideration of combined effects of investment strategies, policy modifications, etc, which were either impossible or difficult tasks using traditional approaches. (3) It gives a shortcut to the derivation of intermediate solution equations. In addition to the efficiency, the new approach also leads to a standardized procedure to cope with various situations. The thesis covers a wide range of ruin-related and financial topics while developing the unifying stochastic approach. Not only does it attempt to provide insights into the unification of quantities in ruin theory, the thesis also seeks to extend its applications in other related areas.

As ruin theory evolves in recent years, there has been a variety of quantities pertaining to an insurer's bankruptcy at the centre of focus in the literature. Despite the fact that these quantities are distinct from each other, it was brought to our attention that many solution methods apply to nearly all ruin-related quantities. Such a peculiar similarity among their solution methods inspired us to search for a general form that reconciles those seemingly different ruin-related quantities. The stochastic approach proposed in the thesis addresses such issues and contributes to the current literature in three major directions. (1) It provides a new function that unifies many existing ruin-related quantities and that produces more new quantities of potential use in both practice and academia. (2) It applies generally to a vast majority of risk processes and permits the consideration of combined effects of investment strategies, policy modifications, etc, which were either impossible or difficult tasks using traditional approaches. (3) It gives a shortcut to the derivation of intermediate solution equations. In addition to the efficiency, the new approach also leads to a standardized procedure to cope with various situations. The thesis covers a wide range of ruin-related and financial topics while developing the unifying stochastic approach. Not only does it attempt to provide insights into the unification of quantities in ruin theory, the thesis also seeks to extend its applications in other related areas.

We prove that every (nice) Feller process is an It^o process in the sense of Cinlar, Jacod, Protter and Sharpe (1980). Next we generalize the notion of the symbol and define it for this larger class of processes. As examples the solutions of stochastic differential equations are considered. The symbol is then used to derive a quick approach to the semimartingale characteristics as well as the generator of the process under consideration. Finally we give some examples of how our methods work for processes used in mathematical finance. / Wir haben gezeigt, dass jeder (nette) Feller Prozess ein It^o Prozess im Sinne von Cinlar, Jacod, Protter und Sharpe (1980) ist. Es stellt sich heraus, dass man den Begriff des Symbols, der für Feller Prozesse bekannt ist, auf diese größere Klasse verallgemeinern kann. Dieses Symbol haben wir für die Lösungen verschiedener stochastischer Differentialgleichungen berechnet. Außerdem haben wir gezeigt, dass das Symbol einen schnellen Zugang zur Berechnung der Semimartingal-Charakteristiken und des Erzeugers eines It^o Prozesses liefert. Zuletzt wurden die Ergebnisse auf Prozesse angewendet, die in der Finanzmathematik gebräuchlich sind. - (Die Dissertation ist veröffentlicht im Shaker Verlag GmbH, Postfach 101818, 52018 Aachen, Deutschland, http://www.shaker.de, ISBN: 978-3-8322-8244-8)

In this thesis, nonlinearly perturbed stochastic models in discrete time are considered. We give algorithms for construction of asymptotic expansions with respect to the perturbation parameter for various quantities of interest. In particular, asymptotic expansions are given for solutions of renewal equations, quasi-stationary distributions for semi-Markov processes, and ruin probabilities for risk processes. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 4: Manuscript. Paper 5: Manuscript. Paper 6: Manuscript.</p>

Dans le cadre de l'optimisation de la fiabilité, Thales Optronique intègre désormais dans ses équipements, des systèmes d'observation de leur état de fonctionnement. Cette fonction est réalisée par des HUMS (Health & Usage Monitoring System). L'objectif de cette thèse est de mettre en place dans le HUMS, un programme capable d'évaluer l'état du système, de détecter les dérives de fonctionnement, d'optimiser les opérations de maintenance et d'évaluer les risques d'échec d'une mission, en combinant les procédés de traitement des données opérationnelles (collectées sur chaque appareil grâce au HUMS) et prévisionnelles (issues des analyses de fiabilité et des coûts de maintenance, de réparation et d'immobilisation). Trois algorithmes ont été développés. Le premier, basé sur un modèle de chaînes de Markov cachées, permet à partir de données opérationnelles, d'estimer à chaque instant l'état du système, et ainsi, de détecter un mode de fonctionnement dégradé de l'équipement (diagnostic). Le deuxième algorithme permet de proposer une stratégie de maintenance optimale et dynamique. Il consiste à rechercher le meilleur instant pour réaliser une maintenance, en fonction de l'état estimé de l'équipement. Cet algorithme s'appuie sur une modélisation du système, par un processus Markovien déterministe par morceaux (noté PDMP) et sur l'utilisation du principe d'arrêt optimal. La date de maintenance est déterminée à partir des données opérationnelles, prévisionnelles et de l'état estimé du système (pronostic). Quant au troisième algorithme, il consiste à déterminer un risque d'échec de mission et permet de comparer les risques encourus suivant la politique de maintenance choisie.Ce travail de recherche, développé à partir d'outils sophistiqués de probabilités théoriques et numériques, a permis de définir un protocole de maintenance conditionnelle à l'état estimé du système, afin d'améliorer la stratégie de maintenance, la disponibilité des équipements au meilleur coût, la satisfaction des clients et de réduire les coûts d'exploitation. / As part of optimizing the reliability, Thales Optronics now includes systems that examine the state of its equipment. This function is performed by HUMS (Health & Usage Monitoring System). The aim of this thesis is to implement in the HUMS a program based on observations that can determine the state of the system, anticipate and alert about the excesses of operation, optimize maintenance operations and evaluate the failure risk of a mission, by combining treatment processes of operational data (collected on each equipment thanks to HUMS) and predictive data (resulting from reliability analysis and cost of maintenance, repair and standstill). Three algorithms have been developed. The first, based on hidden Markov model, allows to estimate at each time the state of the system from operational data, and thus, to detect a degraded mode of equipment (diagnostic). The second algorithm is used to propose an optimal and dynamic maintenance strategy. We want to estimate the best time to perform maintenance, according to the estimated state of equipment. This algorithm is based on a system modeling by a piecewise deterministic Markov process (noted PDMP) and the use of the principle of optimal stopping.The maintenance date is determined from operational and predictive data and the estimated state of the system (prognosis). The third algorithm determines the failure risk of a mission and compares risks following the chosen maintenance policy.This research, developed from sophisticated tools of theoretical and numerical probabilities, allows us to define a maintenance policy adapted to the state of the system, to improve maintenance strategy, the availability of equipment at the lowest cost, customer satisfaction, and reduce operating costs.

La thèse porte sur la conception et l'analyse mathématique de méthodes de Monte Carlo fiables et précises pour l'estimation de la (très petite) probabilité qu'un processus de Markov atteigne une région critique de l'espace d'état avant un instant final déterministe. L'idée sous-jacente aux méthodes de branchement multi-niveaux étudiées ici est de mettre en place une suite emboitée de régions intermédiaires de plus en plus critiques, de telle sorte qu'atteindre une région intermédiaire donnée sachant que la région intermédiaire précédente a déjà été atteinte, n'est pas si rare. En pratique, les trajectoires sont propagées, sélectionnées et répliquées dès que la région intermédiaire suivante est atteinte, et il est facile d'estimer avec précision la probabilité de transition entre deux régions intermédiaires successives. Le biais dû à la discrétisation temporelle des trajectoires du processus de Markov est corrigé en utilisant des régions intermédiaires perturbées, comme proposé par Gobet et Menozzi. Une version adaptative consiste à définir automatiquement les régions intermédiaires, à l’aide de quantiles empiriques. Néanmoins, une fois que le seuil a été fixé, il est souvent difficile voire impossible de se rappeler où (dans quel état) et quand (à quel instant) les trajectoires ont dépassé ce seuil pour la première fois, le cas échéant. La contribution de la thèse consiste à utiliser une première population de trajectoires pilotes pour définir le prochain seuil, à utiliser une deuxième population de trajectoires pour estimer la probabilité de dépassement du seuil ainsi fixé, et à itérer ces deux étapes (définition du prochain seuil, et évaluation de la probabilité de transition) jusqu'à ce que la région critique soit finalement atteinte. La convergence de cet algorithme adaptatif à deux étapes est analysée dans le cadre asymptotique d'un grand nombre de trajectoires. Idéalement, les régions intermédiaires doivent êtres définies en terme des variables spatiale et temporelle conjointement (par exemple, comme l'ensemble des états et des temps pour lesquels une fonction scalaire de l’état dépasse un niveau intermédiaire dépendant du temps). Le point de vue alternatif proposé dans la thèse est de conserver des régions intermédiaires simples, définies en terme de la variable spatiale seulement, et de faire en sorte que les trajectoires qui dépassent un seuil précocement sont davantage répliquées que les trajectoires qui dépassent ce même seuil plus tardivement. L'algorithme résultant combine les points de vue de l'échantillonnage pondéré et du branchement multi-niveaux. Sa performance est évaluée dans le cadre asymptotique d'un grand nombre de trajectoires, et en particulier un théorème central limite est obtenu pour l'erreur d'approximation relative. / The thesis deals with the design and mathematical analysis of reliable and accurate Monte Carlo methods in order to estimate the (very small) probability that a Markov process reaches a critical region of the state space before a deterministic final time. The underlying idea behind the multilevel splitting methods studied here is to design an embedded sequence of intermediate more and more critical regions, in such a way that reaching an intermediate region, given that the previous intermediate region has already been reached, is not so rare. In practice, trajectories are propagated, selected and replicated as soon as the next intermediate region is reached, and it is easy to accurately estimate the transition probability between two successive intermediate regions. The bias due to time discretization of the Markov process trajectories is corrected using perturbed intermediate regions as proposed by Gobet and Menozzi. An adaptive version would consist in the automatic design of the intermediate regions, using empirical quantiles. However, it is often difficult if not impossible to remember where (in which state) and when (at which time instant) did each successful trajectory reach the empirically defined intermediate region. The contribution of the thesis consists in using a first population of pilot trajectories to define the next threshold, in using a second population of trajectories to estimate the probability of exceeding this empirically defined threshold, and in iterating these two steps (definition of the next threshold, and evaluation of the transition probability) until the critical region is reached. The convergence of this adaptive two-step algorithm is studied in the asymptotic framework of a large number of trajectories. Ideally, the intermediate regions should be defined in terms of the spatial and temporal variables jointly (for example, as the set of states and times for which a scalar function of the state exceeds a time-dependent threshold). The alternate point of view proposed in the thesis is to keep intermediate regions as simple as possible, defined in terms of the spatial variable only, and to make sure that trajectories that manage to exceed a threshold at an early time instant are more replicated than trajectories that exceed the same threshold at a later time instant. The resulting algorithm combines importance sampling and multilevel splitting. Its preformance is evaluated in the asymptotic framework of a large number of trajectories, and in particular a central limit theorem is obtained for the relative approximation error.