This thesis, consisting of six papers and a summary, studies the area of continuous time financial mathematics. A unifying theme for many of the problems studied is the implications of possible mis-specifications of models. Intimately connected with this question is, perhaps surprisingly, convexity properties of option prices. We also study qualitative behavior of different optimal stopping boundaries appearing in option pricing. In Paper I a new condition on the contract function of an American option is provided under which the option price increases monotonically in the volatility. It is also shown that American option prices are continuous in the volatility. In Paper II an explicit pricing formula for the perpetual American put option in the Constant Elasticity of Variance model is derived. Moreover, different properties of this price are studied. Paper III deals with the Russian option with a finite time horizon. It is shown that the value of the Russian option solves a certain free boundary problem. This information is used to analyze the optimal stopping boundary. A study of perpetual game options is performed in Paper IV. One of the main results provides a condition under which the value of the option is increasing in the volatility. In Paper V options written on several underlying assets are considered. It is shown that, within a large class of models, the only model for the stock prices that assigns convex option prices to all convex contract functions is geometric Brownian motion. Finally, in Paper VI it is shown that the optimal stopping boundary for the American put option is convex in the standard Black-Scholes model.

In this work we address to the problems of the loan origination decision-making systems. In accordance with the basic principles of the loan origination process we considered the main rules of a clients parameters estimation, a change-point problem for the given data and a disorder moment detection problem for the real-time observations. In the first part of the work the main principles of the parameters estimation are given. Also the change-point problem is considered for the given sample in the discrete and continuous time with using the Maximum likelihood method. In the second part of the work the disorder moment detection problem for the real-time observations is considered as a disorder problem for a non-homogeneous Poisson process. The corresponding optimal stopping problem is reduced to the free-boundary problem with a complete analytical solution for the case when the intensity of defaults increases. Thereafter a scheme of the real time detection of a disorder moment is given.

在生態學與生物學上，物種數常作為生物多樣性的指標，以估計單一群體物種數為例，較知名的方法首推Good (1953)以在樣本中出現一次的物種為基礎，提出的物種數估計方法堪稱的先驅，隨後許多文獻延伸Good的想法，發展出許多的估計方法，例如Burham and Overton (1978)的摺刀估計法，Chao and Lee (1992)則以涵蓋機率方式估計。相對而言，兩群體的共有物種數的研究少有人探討，目前以Chao et al. (2000)的估計式較為知名。 本研究參考Good (1953)提出估計未發現物種出現機率的想法，估計未發現共有物種的機率，並以Burham and Overton (1978)中應用摺刀法估計物種數的概念，建立一階摺刀估計式與變異數，且另行以多項分配公式推導變異數估計式，進行電腦模擬與實際資料驗證並與Chao et al. (2000)提出的共有物種估計式比較。最後根據Rasmussen and Starr (1979)以抽樣成本建立最適停止規則的概念，應用於本研究所提出的估計式，並經由電腦模擬找出抽樣成本與物種分佈均勻程度的關聯，可作為設定停止規則的依據。 / The number of species is often used to measure the biodiversity of a population in ecology and biology. Good (1953) proposed a famous estimate for the number of species based on the probability of unseen species. Subsequently, many studies applied Good’s idea to create new estimation methods, For example, the Jackknife estimate by Burham and Overton (1978), and the estimate by using the sample coverage probability in Chao and Lee (1992) are two famous examples. However, not many studies focus on estimating the number of shared species of two populations, except the method by Chao et al. (2000). In this study, we modify Good’s idea and extend the Jackknife method of Burham and Overton (1978) to develop the estimate for the number of shared species of two populations. In addition, we also establish the variance formula of the estimator by using the multinomial distribution. Subsequently, we use computer simulation and real data sets to evaluate the proposed method, and compare them with the estimator by Chao et al. (2000). Finally, we adapt the idea of optimal stopping rule by Rasmussen and Starr (1979) and combine it with the proposed jackknife estimate. We found that using the sampling cost as the stopping rule is a feasible approach for estimating the number of shared species.

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This thesis consists of a summary and five papers, dealing with financial applications of optimal stopping, optimal control and volatility. In Paper I, we present a method to recover a time-independent piecewise constant volatility from a finite set of perpetual American put option prices. In Paper II, we study the optimal liquidation problem under the assumption that the asset price follows a geometric Brownian motion with unknown drift, which takes one of two given values. The optimal strategy is to liquidate the first time the asset price falls below a monotonically increasing, continuous time-dependent boundary. In Paper III, we investigate the optimal liquidation problem under the assumption that the asset price follows a jump-diffusion with unknown intensity, which takes one of two given values. The best liquidation strategy is to sell the asset the first time the jump process falls below or goes above a monotone time-dependent boundary. Paper IV treats the optimal dividend problem in a model allowing for positive jumps of the underlying firm value. The optimal dividend strategy is of barrier type, i.e. to pay out all surplus above a certain level as dividends, and then pay nothing as long as the firm value is below this level. Finally, in Paper V it is shown that a necessary and sufficient condition for the explosion of implied volatility near expiry in exponential Lévy models is the existence of jumps towards the strike price in the underlying process.

<p>Diss. Stockholm : Handelshögskolan, 2013. Sammanfattning jämte 3 uppsatser.</p>

Cette thèse est consacrée à l’analyse de ces modèles particulaires pour les mathématiques financières.Le manuscrit est organisé en quatre chapitres. Chacun peut être lu séparément.Le premier chapitre présente le travail de thèse de manière globale, définit les objectifs et résume les principales contributions. Le deuxième chapitre constitue une introduction générale à la théorie des méthodes particulaire, et propose un aperçu de ses applications aux mathématiques financières. Nous passons en revue les techniques et les résultats principaux sur les systèmes de particules en interaction, et nous expliquons comment ils peuvent être appliques à la solution numérique d’une grande variété d’applications financières, telles que l’évaluation d’options compliquées qui dépendent des trajectoires, le calcul de sensibilités, l’évaluation d’options américaines ou la résolution numérique de problèmes de contrôle et d’estimation avec observation partielle.L’évaluation d’options américaines repose sur la résolution d’une équation d’évolution à rebours, nommée l’enveloppe de Snell dans la théorie du contrôle stochastique et de l’arrêt optimal. Les deuxième et troisième chapitres se concentrent sur l’analyse de l’enveloppe de Snell et de ses extensions à différents cas particuliers. Un ensemble de modèles particulaires est alors proposé et analysé numériquement. / This thesis is concerned with the analysis of these particle models for computational finance.The manuscript is organized in four chapters. Each of them could be read separately.The first chapter provides an overview of the thesis, outlines the motivation and summarizes the major contributions. The second chapter gives a general in- troduction to the theory of interacting particle methods, with an overview of their applications to computational finance. We survey the main techniques and results on interacting particle systems and explain how they can be applied to the numerical solution of a variety of financial applications; to name a few: pricing complex path dependent European options, computing sensitivities, pricing American options, as well as numerically solving partially observed control and estimation problems.The pricing of American options relies on solving a backward evolution equation, termed Snell envelope in stochastic control and optimal stopping theory. The third and fourth chapters focus on the analysis of the Snell envelope and its variation to several particular cases. Different type of particle models are proposed and studied.

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.

Our vision is as yet unsurpassed by machines because of the sophisticated representations of objects in our brains. This representation is vastly different from a pixel-based representation used in machine storages. It is this sophisticated representation that enables us to perceive two faces as very different, i.e, they are far apart in the “perceptual space”, even though they are close to each other in their pixel-based representations. Neuroscientists have proposed distances between responses of neurons to the images (as measured in macaque monkeys) as a quantification of the “perceptual distance” between the images. Let us call these neuronal dissimilarity indices of perceptual distances. They have also proposed behavioural experiments to quantify these perceptual distances. Human subjects are asked to identify, as quickly as possible, an oddball image embedded among multiple distractor images. The reciprocal of the search times for identifying the oddball is taken as a measure of perceptual distance between the oddball and the distractor. Let us call such estimates as behavioural dissimilarity indices. In this thesis, we describe a decision-theoretic model for visual search that suggests a connection between these two notions of perceptual distances. In the first part of the thesis, we model visual search as an active sequential hypothesis testing problem. Our analysis suggests an appropriate neuronal dissimilarity index which correlates strongly with the reciprocal of search times. We also consider a number of alternative possibilities such as relative entropy (Kullback-Leibler divergence), the Chernoff entropy and the L1-distance associated with the neuronal firing rate profiles. We then come up with a means to rank the various neuronal dissimilarity indices based on how well they explain the behavioural observations. Our proposed dissimilarity index does better than the other three, followed by relative entropy, then Chernoff entropy and then L1 distance. In the second part of the thesis, we consider a scenario where the subject has to find an oddball image, but without any prior knowledge of the oddball and distractor images. Equivalently, in the neuronal space, the task for the decision maker is to find the image that elicits firing rates different from the others. Here, the decision maker has to “learn” the underlying statistics and then make a decision on the oddball. We model this scenario as one of detecting an odd Poisson point process having a rate different from the common rate of the others. The revised model suggests a new neuronal dissimilarity index. The new dissimilarity index is also strongly correlated with the behavioural data. However, the new dissimilarity index performs worse than the dissimilarity index proposed in the first part on existing behavioural data. The degradation in performance may be attributed to the experimental setup used for the current behavioural tasks, where search tasks associated with a given image pair were sequenced one after another, thereby possibly cueing the subject about the upcoming image pair, and thus violating the assumption of this part on the lack of prior knowledge of the image pairs to the decision maker. In conclusion, the thesis provides a framework for connecting the perceptual distances in the neuronal and the behavioural spaces. Our framework can possibly be used to analyze the connection between the neuronal space and the behavioural space for various other behavioural tasks.

Local electricity markets can be defined broadly as 'future electricity market designs involving domestic customers, demand-side response and energy storage'. Like current deregulated electricity markets, these localised derivations present specific stochastic optimisation problems in which the dynamic and random nature of the market is intertwined with the physical needs of its participants. Moreover, the types of contracts and constraints in this setting are such that 'games' naturally emerge between the agents. Advanced modelling techniques beyond classical mathematical finance are therefore key to their analysis. This thesis aims to study contracts in these local electricity markets using the mathematical theories of stochastic optimal control and games. Chapter 1 motivates the research, provides an overview of the electricity market in Great Britain, and summarises the content of this thesis. It introduces three problems which are studied later in the thesis: a simple control problem involving demand-side management for domestic customers, and two examples of games within local electricity markets, one of them involving energy storage. Chapter 2 then reviews the literature most relevant to the topics discussed in this work. Chapter 3 investigates how electric space heating loads can be made responsive to time varying prices in an electricity spot market. The problem is formulated mathematically within the framework of deterministic optimal control, and is analysed using methods such as Pontryagin's Maximum Principle and Dynamic Programming. Numerical simulations are provided to illustrate how the control strategies perform on real market data. The problem of Chapter 3 is reformulated in Chapter 4 as one of optimal switching in discrete-time. A martingale approach is used to establish the existence of an optimal strategy in a very general setup, and also provides an algorithm for computing the value function and the optimal strategy. The theory is exemplified by a numerical example for the motivating problem. Chapter 5 then continues the study of finite horizon optimal switching problems, but in continuous time. It also uses martingale methods to prove the existence of an optimal strategy in a fairly general model. Chapter 6 introduces a mathematical model for a game contingent claim between an electricity supplier and generator described in the introduction. A theory for using optimal switching to solve such games is developed and subsequently evidenced by a numerical example. An optimal switching formulation of the aforementioned game contingent claim is provided for an abstract Markovian model of the electricity market. The final chapter studies a balancing services contract between an electricity transmission system operator (SO) and the owner of an electric energy storage device (battery operator or BO). The objectives of the SO and BO are combined in a non-zero sum stochastic differential game where one player (BO) uses a classic control with continuous effects, whereas the other player (SO) uses an impulse control (discontinuous effects). A verification theorem proving the existence of Nash equilibria in this game is obtained by recursion on the solutions to Hamilton-Jacobi-Bellman variational PDEs associated with non-zero sum controller-stopper games.