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Dissertations / Theses on the topic 'Optimal Stopping Theory'
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Qiang, Li. "Pair Trading in Optimal Stopping Theory." Thesis, Uppsala University, Department of Mathematics, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-119421.
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Jones, Martin Lee. "Universal constants in optimal stopping theory." Diss., Georgia Institute of Technology, 1989. http://hdl.handle.net/1853/30092.
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Landon, Nicolas. "Almost sure optimal stopping times : theory and applications." Phd thesis, Ecole Polytechnique X, 2013. http://pastel.archives-ouvertes.fr/pastel-00788067.
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Résumé : Cette thèse comporte 8 chapitres. Le chapitre 1 est une introduction aux problématiques rencontrées sur les marchés énergétiques : fréquence d'intervention faible, coûts de transaction élevés, évaluation des options spread. Le chapitre 2 étudie la convergence de l'erreur de couverture d'une option call dans le modèle de Bachelier, pour des coûts de transaction proportionnels (modèle de Leland-Lott) et lorsque la fréquence d'intervention devient infinie. Il est prouvé que cette erreur est bornée par une variable aléatoire proportionnelle au taux de transaction. Cependant, les démonstrations de convergence en probabilité demandent des régularités sur les sensibilités assez restrictives en pratique. Les chapitres suivants contournent ces obstacles en étudiant des convergences presque sûres. Le chapitre 3 développe tout d'abord de nouveaux outils de convergence presque sûre. Ces résultats ont de nombreuses conséquences sur le contrôle presque sûr de martingales et de leur variation quadratique, ainsi que de leurs incréments entre deux temps d'arrêt généraux. Ces résultats de convergence trajectorielle sont connus pour être difficiles à obtenir sans information sur les lois. Par la suite, nous appliquons ces résultats à la minimisation presque sûre de la variation quadratique renormalisée de l'erreur de couverture d'une option de payoff général (cadre multidimensionnel, payoff asiatique, lookback) sur une large classe de temps d'intervention. Une borne inférieure à notre critère est trouvée et une suite minimisante de temps d'arrêt optimale est exhibée : il s'agit de temps d'atteinte d'ellipsoïde aléatoire, dépendant du gamma de l'option. Le chapitre 4 étudie la convergence de l'erreur de couverture d'une option de payoff convexe (dimension 1) en prenant en compte des coûts de transaction à la Leland-Lott. Nous décomposons l'erreur de couverture en une partie martingale et une partie négligeable, puis nous minimisons la variation quadratique de cette martingale sur une classe de temps d'atteintes générales pour des Deltas vérifiant une certaine EDP non-linéaire sur les dérivées secondes. Nous exhibons aussi une suite de temps d'arrêt atteignant cette borne. Des tests numériques illustrent notre approche par rapport à une série de stratégies connues de la littérature. Le chapitre 5 étend le chapitre 3 en considérant une fonctionnelle des variations discrètes d'ordre Y et de Z de deux processus d'Itô Y et Z à valeurs réelles, la minimisation étant sur une large classe de temps d'arrêt servant au calcul des variations discrètes. Borne inférieure et suite minimisant sont obtenues. Une étude numérique sur les coûts de transaction est faite. Le chapitre 6 étudie la discrétisation d'Euler d'un processus multidimensionnel X dirigé par une semi-martingale d'Itô Y . Nous minimisons sur les temps de la grille de discrétisation un critère quadratique sur l'erreur du schéma. Nous trouvons une borne inférieure et une grille optimale, ne dépendant que des données observables. Le chapitre 7 donne un théorème limite centrale pour des discrétisations d'intégrale stochastique sur des grilles de temps d'atteinte d'ellipsoïdes adaptées quelconque. La corrélation limite est conséquence d'asymptotiques fins sur les problèmes de Dirichlet. Dans le chapitre 8, nous nous intéressons aux formules d'expansion pour les options sur spread, pour des modèles à volatilité locale. La clé de l'approche consiste à conserver la propriété de martingale de la moyenne arithmétique et à exploiter la structure du payoff call. Les tests numériques montrent la pertinence de l'approche.
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Christensen, Sören [Verfasser]. "Contributions to the theory of optimal stopping / Sören Christensen." Kiel : Universitätsbibliothek Kiel, 2010. http://d-nb.info/1020001488/34.
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Oryu, Tadao. "An Excursion-Theoretic Approach to Optimal Stopping Problems." 京都大学 (Kyoto University), 2017. http://hdl.handle.net/2433/225370.
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Bergström, Jonas. "Pricing the American Option Using Itô’s Formula and Optimal Stopping Theory." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-217176.
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Islas, Anguiano Jose Angel. "Optimal Strategies for Stopping Near the Top of a Sequence." Thesis, University of North Texas, 2015. https://digital.library.unt.edu/ark:/67531/metadc822812/.
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In Chapter 1 the classical secretary problem is introduced. Chapters 2 and 3 are variations of this problem. Chapter 2, discusses the problem of maximizing the probability of stopping with one of the two highest values in a Bernoulli random walk with arbitrary parameter p and finite time horizon n. The optimal strategy (continue or stop) depends on a sequence of threshold values (critical probabilities) which has an oscillating pattern. Several properties of this sequence have been proved by Dr. Allaart. Further properties have been recently proved. In Chapter 3, a gambler will observe a finite sequence of continuous random variables. After he observes a value he must decide to stop or continue taking observations. He can play two different games A) Win at the maximum or B) Win within a proportion of the maximum. In the first section the sequence to be observed is independent. It is shown that for each n>1, theoptimal win probability in game A is bounded below by (1-1/n)^{n-1}. It is accomplished by reducing the problem to that of choosing the maximum of a special sequence of two-valued random variables and applying the sum-the-odds theorem of Bruss (2000). Secondly, it is assumed the sequence is i.i.d. The best lower bounds are provided for the winning probabilities in game B given any continuous distribution. These bounds are the optimal win probabilities of a game A which was examined by Gilbert and Mosteller (1966).
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Vaicenavicius, Juozas. "Optimal Sequential Decisions in Hidden-State Models." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-320809.
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This doctoral thesis consists of five research articles on the general topic of optimal decision making under uncertainty in a Bayesian framework. The papers are preceded by three introductory chapters. Papers I and II are dedicated to the problem of finding an optimal stopping strategy to liquidate an asset with unknown drift. In Paper I, the price is modelled by the classical Black-Scholes model with unknown drift. The first passage time of the posterior mean below a monotone boundary is shown to be optimal. The boundary is characterised as the unique solution to a nonlinear integral equation. Paper II solves the same optimal liquidation problem, but in a more general model with stochastic regime-switching volatility. An optimal liquidation strategy and various structural properties of the problem are determined. In Paper III, the problem of sequentially testing the sign of the drift of an arithmetic Brownian motion with the 0-1 loss function and a constant cost of observation per unit of time is studied from a Bayesian perspective. Optimal decision strategies for arbitrary prior distributions are determined and investigated. The strategies consist of two monotone stopping boundaries, which we characterise in terms of integral equations. In Paper IV, the problem of stopping a Brownian bridge with an unknown pinning point to maximise the expected value at the stopping time is studied. Besides a few general properties established, structural properties of an optimal strategy are shown to be sensitive to the prior. A general condition for a one-sided optimal stopping region is provided. Paper V deals with the problem of detecting a drift change of a Brownian motion under various extensions of the classical Wiener disorder problem. Monotonicity properties of the solution with respect to various model parameters are studied. Also, effects of a possible misspecification of the underlying model are explored.
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Allen, Andrew. "A Random Walk Version of Robbins' Problem." Thesis, University of North Texas, 2018. https://digital.library.unt.edu/ark:/67531/metadc1404568/.
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Robbins' problem is an optimal stopping problem where one seeks to minimize the expected rank of their observations among all observations. We examine random walk analogs to Robbins' problem in both discrete and continuous time. In discrete time, we consider full information and relative ranks versions of this problem. For three step walks, we give the optimal stopping rule and the expected rank for both versions. We also give asymptotic upper bounds for the expected rank in discrete time. Finally, we give upper and lower bounds for the expected rank in continuous time, and we show that the expected rank in the continuous time problem is at least as large as the normalized asymptotic expected rank in the full information discrete time version.
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Dyrssen, Hannah. "Valuation and Optimal Strategies in Markets Experiencing Shocks." Doctoral thesis, Uppsala universitet, Tillämpad matematik och statistik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-316578.
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This thesis treats a range of stochastic methods with various applications, most notably in finance. It is comprised of five articles, and a summary of the key concepts and results these are built on. The first two papers consider a jump-to-default model, which is a model where some quantity, e.g. the price of a financial asset, is represented by a stochastic process which has continuous sample paths except for the possibility of a sudden drop to zero. In Paper I prices of European-type options in this model are studied together with the partial integro-differential equation that characterizes the price. In Paper II the price of a perpetual American put option in the same model is found in terms of explicit formulas. Both papers also study the parameter monotonicity and convexity properties of the option prices. The third and fourth articles both deal with valuation problems in a jump-diffusion model. Paper III concerns the optimal level at which to exercise an American put option with finite time horizon. More specifically, the integral equation that characterizes the optimal boundary is studied. In Paper IV we consider a stochastic game between two players and determine the optimal value and exercise strategy using an iterative technique. Paper V employs a similar iterative method to solve the statistical problem of determining the unknown drift of a stochastic process, where not only running time but also each observation of the process is costly.
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Lindensjö, Kristoffer. "Essays in financial mathematics." Doctoral thesis, Handelshögskolan i Stockholm, Institutionen för Finansiell ekonomi, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:hhs:diva-2145.
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黃嘉彥. "A Fast mode decision Algorithm based on Optimal Stopping Theory for SVC Multi-Layer Encoder Control." Thesis, 2011. http://ndltd.ncl.edu.tw/handle/67677002190431362356.
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碩士<br>國立交通大學<br>資訊科學與工程研究所<br>100<br>This thesis proposed a fast mode decision algorithm for scalable video coding (SVC) multi-layer encoder control (MLEC). The proposed algorithm attempts to improve the problem of the fast algorithm proposed by Zhao’s et al. based on the optimal stopping theory (OST), in which the problem is the benefit obtained of the different mode is regarded as the same and only the distance considered between the stopping point and the position of the next relative best mode in the mode sequence. Unfortunately, this distance cannot guarantee the amount of RD-cost decrease in the next relative best mode; additionally, the amount of required computational complexity to the next relative best mode cannot be anticipated either. Thus, this thesis will take aforementioned points into consideration in order to create a more efficient fast mode decision algorithm.
This thesis analyzed the stopping point in the mode pair sequence, which explore the amount of RD-cost decrease by finding the next relative best mode pair, and analyze the different amount of computational complexity between the different mode pairs. A table containing the above two points will also be provided for calculation of the optimal stopping point. Additionally, this thesis will proposed a more refined encoding method by utilizing statistics and data analysis to identify inferior encoding mode pairs and then refining them. Experimental result demonstrates that the proposed fast mode decision algorithm compares to the method of exhaustive search can achieve time saving about 80% to 85% with negligible quality degradation. Furthermore, the proposed method is superior to those of the other two types because of its stability and encoding quality.
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"Optimal stopping of Markov processes : Hilbert space theory, approximation algorithms, and an application to pricing high-dimensional financial derivatives." Massachusetts Institute of Technology, Laboratory for Information and Decision Systems, 1997. http://hdl.handle.net/1721.1/3454.
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Snyder, Lukas. "Cost minimization under sequential testing procedures using a Bayesian approach." 2013. http://liblink.bsu.edu/uhtbin/catkey/1712476.
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In sequential testing an observer must choose when to observe additional data points and when to stop observation and make a decision. This stopping rule is traditionally based upon probability of error as well as certain cost parameters. The proposed stopping rule will instruct the observer to cease observation once the expected cost of the next observation increases. There is often a great deal of information about what the observer should see. This information will be used to develop a prior distribution for the parameters. The proposed stopping rule will be analyzed and compared to other stopping rules. Analysis of simulated data shows under which conditions the cost of the proposed stopping rule will approximate the minimum expected cost.<br>Department of Mathematical Sciences
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Vaidhiyan, Nidhin Koshy. "Neuronal Dissimilarity Indices that Predict Oddball Detection in Behaviour." Thesis, 2016. http://etd.iisc.ac.in/handle/2005/2669.
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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.
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Vaidhiyan, Nidhin Koshy. "Neuronal Dissimilarity Indices that Predict Oddball Detection in Behaviour." Thesis, 2016. http://etd.iisc.ernet.in/handle/2005/2669.
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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.
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Jürgens, Stephan [Verfasser]. "Beiträge zur Theorie des mehrfachen optimalen Stoppens / vorgelegt von Stephan Jürgens." 2010. http://d-nb.info/1010561359/34.
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DE, ANGELIS SIMONE. "A singular stochastic control problem for hydropower generation in renewable energy markets." Doctoral thesis, 2018. http://hdl.handle.net/11573/1211197.
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We consider a singular stochastic control problem for hydroelectric power production in an energy market where the electricity spot market prices dynamics is described by a Vasicek's process, allowing also for negative prices.
We propose a hydroelectric production system that can react in two different modes when it is convenient to produce energy through an instantaneous release of water. We endow the system with the possibility of producing ``less efficiently'' when negative prices appear in the market, but it is still preferable to produce instantaneously rather than waiting for positive prices.
We defined a novel optimization problem whose performance functional exhibits a state-dependent instantaneous marginal revenue whose sign is directly affected by the sign of the prices dynamics.
We aim to maximize such functional among all the admissible control policies into the class of the adapted stochastic processes whose paths are not necessarily absolutely continuous with respect to the Lebesgue's measure but only non-decreasing, left-continuous and with finite right limits (càglàd).
We prove the Verification Theorem, allowing to characterize the value function of our singular stochastic control problem among the solutions of the associated Hamilton-Jacobi-Bellman equation which turns to be a variational inequality with state-dependent gradient constraint. The Verification Theorem sheds light on the structure of the optimal control which turns to be a purely discontinuous process that, at the first time of action, exerts all the available fuel with a single instantaneous jump.
Under some assumptions on the characteristics of our hydropower production model, we identify the value function of the optimal control problem in terms of the optimal reward function of an associated family of optimal stopping problems. We identified a unique positive boundary, separating the action and inaction regions and we showed that the optimal strategy consists in completely discharge the water reservoir as soon as the price dynamics reaches values grater or equal such optimal threshold.
We highlight the difficulties that arise when the aforementioned assumptions are replaced by other alternative hypotheses. In this more challenging context, we provide some intuitions on the tricky structure of the action and inaction regions as well as on the nature of the candidate optimal control policy.