Cette thèse est composée de quatre chapitres. Le premier chapitre traite de l'évaluation de produits financiers dans un modèle comportant un saut pour l'actif risque. Ce saut représente la faillite de l'entreprise correspondante. On étudie alors l'évaluation des prix d'options par indifférence d'utilité dans un cadre d'utilité exponentielle. Par des techniques de programmation dynamique on montre que le prix d'un Bond est solution d'une équation différentielle et le prix d'options dépendantes de l'actif est solution d'une équation aux dérives partielles d'Hamilton-Jacobi-Bellman. Le saut dans la dynamique de l'actif risque induit des différences avec le modèle de Merton que nous tentons de quantifier. Le second chapitre traite d'un marché comportant des sauts : les paris sur le football. Nous rappelons les différentes familles de modèles pour un match de football et introduisons un modèle complet permettant d'évaluer les prix des différents produits apparus sur ce marché ces dix dernières années. La complexité de ce modèle nous amène à étudier un modèle simplifié dont nous étudions les implications et calculons les prix obtenus que l'on compare à la réalité. On remarque que la calibration implicite obtenue génère de très bons résultats en produisant des prix très proches de la réalité. Le troisième chapitre développe le problème de fixation des prix par un teneur de marche monopolistique dans le marché des paris binaires. Ce travail est un prolongement direct au problème introduit par Levitt [Lev04]. Nous généralisons en effet son travail aux cas des paris européens et proposons une méthode pour estimer la méthode de cotation utilisée par le book-maker. Nous montrons que deux hypothèses inextricables peuvent expliquer cette fixation des prix. D'une part, l'incertitude du public sur la vraie valeur ainsi que le caractère extrêmement risque-averse du bookmaker. Le quatrième chapitre prolonge quant à lui cette approche au cas de produits financiers non binaires. Nous examinons différents modèles d'offre et de demande et en déduisons, par des techniques de programmation dynamique, des équations aux dérivées partielles dictant la formation des prix d'achat et de vente. Nous montrons finalement que l'écart entre prix d'achat et prix de vente ne dépend pas de la position du teneur de marche dans l'actif considère. Cependant le prix moyen dépend lui fortement de la quantité détenue par le teneur de marche. Une approche simplifiée est finalement proposée dans le cas multidimensionnel / This thesis is composed of four chapters. The first one deals with the pricing of financial products in a single jump model for the risky asset. This jump represents the bankrupcy of the quoted firm. We study the pricing of derivatives in the context of indifference of utility with an exponential utility. By means of dynamic programming we show that the bond price is solution of an ordinary differential equation and that stock price dependent options are solutions of an equation with partial derivatives of Hamilton-Jacobi-Bellman type generalizing the Black-Scholes one. We then try to quantify differences in the price obtained here and the one from Merton model without jump. The second chapter deals with a specific jump market : the soccer betting market. We recall the different model families for a soccer match and introduce some full model which allows to price the products recently born in this market in last ten years. Nevertheless the model complexity leads us to study a simplified model introduced by Dixon and Robinson from which we are able to derive closed formulas and simulate prices that we compare to market prices. We remark that implicit calibration gives pretty goof fit of market data. Third chapter developps the approach of Levitt [Lev04] on price formation in binary betting market held by a monopolistic market-maker operating in a one time step trading. We generalize Levitt results with european format of betting. We show that prices are distorded on the pressure of demand and offer, that phenomena introducing a market probability that allows to price products under this new measure. We identify some best model for demand and offer and market maker strategy and show that probability change is obvious in case of imperfect information about the value of the product. Fourth chapter generalizes this approach to the case of general payoffs and continuous time. The task is more complex and we just derive partial derivative equations from dynamic programming that enable us to give the bid-ask prices of the product traded by the market-maker. One result is that, in most models, bid-ask spread does not depend on the inventory held by the dealer whereas mid-quote price strongly reflects the unbalance of the dealer

Tesis por compendio / [ES] Los sistemas de drenaje urbano, también conocidos como sistemas de alcantarillado, tienen el propósito de drenar tanto las aguas lluvias como las aguas residuales de nuestras ciudades. Estos sistemas son una de las diferentes infraestructuras civiles que permiten el correcto funcionamiento de las ciudades actuales. Los sistemas de alcantarillados pluviales se encargan de mover las aguas lluvias que caen en la ciudad en forma segura y en un tiempo razonable, sin permitir inundaciones, hacia los cuerpos receptores naturales localizados aguas abajo. Los sistemas de alcantarillado sanitario, por otra parte, tienen el objetivo de recolectar las aguas servidas, de origen domiciliario, industrial y comercial, y trasportarlas en forma segura hacia una planta de tratamiento de aguas residuales antes de enviarlas en forma definitiva hacia un cuerpo receptor o hacia un sistema de reuso, en caso de que este exista. Ambos tipos de alcantarillado, pero particularmente los sanitarios, tienen un profundo impacto sobre la salud pública con implicaciones sobre la viabilidad de una ciudad. La infraestructura de drenaje urbano existente plantea nuevos retos a la ingeniería hidráulica urbana. Por un lado, los sistemas de drenaje de aguas lluvias, además de verse sometidos a problemas de envejecimiento con deterioros de materiales y asentamientos de suelos, se están enfrentando a serios problemas causados por el Cambio Climático. En muchas ciudades este fenómeno está implicando cambios radicales en la hidrología urbana; en muchas ciudades las intensidades y frecuencias de los eventos de lluvia están aumentando con lo cual los sistemas existentes, que fueron diseñados para hidrologías diferentes, se quedan sin capacidad aumentando la frecuencia de las inundaciones urbanas con sus consecuencias sobre la seguridad y salud humana. Por otro lado, los sistemas sanitarios también presentan retos a la ingeniería por problemas de aumento de la densidad poblacional de las ciudades, la falta de resiliencia ante eventos externos como sismos, y a problemas de calidad de agua en los cuerpos receptores al interior de las zonas urbanas, en las aguas freáticas y las corrientes naturales de agua. Estos sistemas también se ven afectados por el deterioro de los materiales y los problemas de asentamiento de los suelos. Ahora, un problema diferente es la no existencia de sistemas de drenaje urbano en muchas ciudades de nuestro mundo, particularmente en aquellas localizadas en países en vías de desarrollo. En la gran mayoría de esos casos, esa falta de alcantarillados es causada por el alto costo de esa infraestructura cuya construcción hace inviable para los gobiernos locales. Además, el crecimiento de las zonas urbanas en esos países agrava el problema. No solamente la infraestructura necesaria se hace más grande y compleja, sino que aumentan los problemas de salud pública asociados con la falta de sanidad ambiental. Por consiguiente, el reto para la ingeniería hidráulica urbana moderna es lograr un mayor acceso a ese servicio esencial. Una de las formas de resolverlo es bajando los costos de construcción y operación de los sistemas de drenaje, haciéndolos financieramente viables a la vez que se mantienen su resiliencia y seguridad. De esta manera se ayudará a cumplir con el Objetivo de Desarrollo Sostenible No. 6 Agua Limpia y Saneamiento. Teniendo en cuenta lo anterior, el objetivo de esta tesis fue proponer una metodología que llevara al diseño de mínimo costo de redes de alcantarillado convencionales, manteniendo su resiliencia y facilidad de operación, a la vez que se cumplieran todas las restricciones hidráulicas, constructivas y de operación que, de acuerdo con la experiencia empírica internacional, son las apropiadas para garantizar un correcto comportamiento del sistema de drenaje. El diseño optimizado de una red de alcantarillado está compuesto por dos partes mutuamente dependientes: la selección del árbol y el diseño hidráulico. En esta tesis se resolvieron / [CA] Els sistemes de drenatge urbà, també coneguts com a sistemes de clavegueram, tenen el propòsit de drenar tant les aigües pluges com les aigües residuals de les nostres ciutats. Aquests sistemes són una de les diferents infraestructures civils que permeten el correcte funcionament de les ciutats actuals. Els sistemes de claveguerams pluvials s'encarreguen de moure les aigües pluges que cauen a la ciutat en forma segura i en un temps raonable, sense permetre inundacions, cap als cossos receptors naturals localitzats aigües avall. Els sistemes de clavegueram sanitari, d'altra banda, tenen l'objectiu de recol·lectar les aigües servides, d'origen domiciliari, industrial i comercial, i transportar-les en forma segura cap a una planta de tractament d'aigües residuals abans d'enviar-les en forma definitiva cap a un cos receptor o cap a un sistema de reuse, en cas que aquest existisca. Tots dos tipus de clavegueram, però particularment els sanitaris, tenen un profund impacte sobre la salut pública amb implicacions sobre la viabilitat d'una ciutat. La infraestructura de drenatge urbà existent planteja nous reptes a l'enginyeria hidràulica urbana. D'una banda, els sistemes de drenatge d'aigües pluges, a més de veure's sotmesos a problemes d'envelliment amb deterioracions de materials i assentaments de sòls, s'estan enfrontant a seriosos problemes causats pel Canvi Climàtic. En moltes ciutats aquest fenomen està implicant canvis radicals en la hidrologia urbana; en moltes ciutats les intensitats i freqüències dels esdeveniments de pluja estan augmentant amb la qual cosa els sistemes existents, que van ser dissenyats per a hidrologies diferents, es queden sense capacitat augmentant la freqüència de les inundacions urbanes amb les seues conseqüències sobre la seguretat i salut humana. D'altra banda, els sistemes sanitaris també presenten reptes a l'enginyeria per problemes d'augment de la densitat poblacional de les ciutats, la falta de resiliència davant esdeveniments externs com a sismes, i a problemes de qualitat d'aigua en els cossos receptors a l'interior de les zones urbanes, en les aigües freàtiques i els corrents naturals d'aigua. Aquests sistemes també es veuen afectats per la deterioració dels materials i els problemes d'assentament dels sòls. Ara, un problema diferent és la no existència de sistemes de drenatge urbà en moltes ciutats del nostre món, particularment en aquelles localitzades en països en vies de desenvolupament. En la gran majoria d'aqueixos casos, aqueixa falta de claveguerams és causada per l'alt cost d'aqueixa infraestructura la construcció de la qual fa inviable per als governs locals. A més, el creixement de les zones urbanes en aqueixos països agreuja el problema. No solament la infraestructura necessària es fa més gran i complexa, sinó que augmenten els problemes de salut pública associats amb la falta de sanitat ambiental. Per consegüent, el repte per a l'enginyeria hidràulica urbana moderna és aconseguir un major accés a aqueix servei essencial. Una de les maneres de resoldre-ho és baixant els costos de construcció i operació dels sistemes de drenatge, fent-los financerament viables alhora que es mantenen la seua resiliència i seguretat. D'aquesta manera s'ajudarà a complir amb l'Objectiu de Desenvolupament Sostenible No. 6 Aigua Neta i Sanejament. Tenint en compte l'anterior, l'objectiu d'aquesta tesi va ser proposar una metodologia que portara al disseny de mínim cost de xarxes de clavegueram convencionals, mantenint la seua resiliència i facilitat d'operació, alhora que es compliren totes les restriccions hidràuliques, constructives i d'operació que, d'acord amb l'experiència empírica internacional, són les apropiades per a garantir un correcte comportament del sistema de drenatge. El disseny optimitzat d'una xarxa de clavegueram està compost per dues parts mútuament dependents: la selecció de l'arbre i el disseny hidràulic. En aquesta tesi es van resoldre els dos problemes en forma separada des del punt / [EN] Urban drainage systems, also known as sewer systems, have the purpose of draining both rainwater and wastewater from our cities. These systems are one of the different civil infrastructures that allow the proper functioning of modern cities. Stormwater systems are responsible for moving the rainwater that falls in the city safely and in a reasonable time, without allowing flooding, to the natural receiving bodies located downstream. On the other hand, wastewater systems are designed to collect sewage from domestic, industrial, and commercial sources and transport it safely to a wastewater treatment plant before sending it definitively to a receiving body or to a reuse system, if available. Both types of sewer systems, but particularly wastewater systems, have an extensive impact on public health with implications for the viability of a city. The existing urban drainage infrastructure presents new challenges to urban hydraulic engineering. On the one hand, stormwater systems, in addition to being subjected to aging problems with deterioration of materials and soil settlement, are facing serious problems caused by climate change. In many cities this phenomenon is implying radical changes in urban hydrology; intensities and frequencies of rainfall events are increasing so that the existing systems, which were designed for different hydrological conditions, are running out of capacity, increasing the frequency of urban flooding with its consequences on safety and human health. On the other hand, wastewater systems also present engineering challenges due to problems of increasing population density in cities, lack of resilience to external events such as earthquakes, and water quality problems in receiving waters within urban areas, in groundwater, and natural water currents. These systems are also affected by the deterioration of materials and soil settlement problems. Now, a different problem is the non-existence of urban drainage systems in many cities of the world, particularly in those located in developing countries. In the vast majority of these cases, the lack of sewers is caused by the high cost of this infrastructure, making its construction unfeasible for local governments. In addition, the growth of urban areas in these countries exacerbates the problem. Not only does the necessary infrastructure become larger and more complex, but the public health problems associated with the lack of environmental sanitation increase. Therefore, the challenge for modern urban water engineering is to achieve greater access to this essential service. One way to solve this is by lowering the construction and operating costs of drainage systems, making them financially viable while maintaining their resilience and safety. This will help meet Sustainable Development Goal No. 6 Clean Water and Sanitation. Considering the above, the aim of this thesis was to propose a methodology that would lead to the minimum cost design of conventional sewer networks, maintaining their resilience and ease of operation, while complying with all the hydraulic, constructive, and operational restrictions that, according to international empirical experience, are appropriate to guarantee a correct behavior of the drainage system. The optimized design of a sewer network is composed of two mutually dependent parts: the layout selection and the hydraulic design. In this thesis, the two problems were solved separately from a mathematical point of view, maintaining their interdependence through an iterative process. For a given layout, which could be random for a first iteration, the hydraulic design problem was solved as a shortest path problem using the Bellman-Ford algorithm that guarantees the global minimum cost for that layout. The problem is model as a directed graph in which the nodes represent the combination of diameters and invert elevations at every manhole, and the arcs represent the diameter and upstream and downstream invert elevation of a specific pi / Saldarriaga, J. (2024). OPTIMAL DESIGN OF URBAN SEWER SYSTEMS [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/202962 / Compendio

[pt] A equação de Isaacs é um exemplo importante de equação elíptica totalmente não-linear, aparecendo em uma grande variedade de disciplinas. Um fato de interesse particular é que tais equações são dirigidas por operadores não convexos. Portanto, são compatíveis com a teoria de EvansKrylov e apresentam delicados desafios quando se trata de sua teoria da regularidade. Descrevemos uma série de resultados recentes sobre a teoria da regularidade da Equação de Isaacs. Estas cobrem estimativas nos espaços Hölder e Sobolev. Argumentamos através de um método genuinamente geométrico, importando informações de uma equação de Bellman relacionada. / [en] Isaacs equation is an important example of fully nonlinear elliptic equation, appearing in a wide of disciplines. Of particular interest is the fact that such equations are driven by nonconvex operators. Therefore, it falls off the scope of the Evans-Krylov theory and poses additional, delicate, challenges when it comes to its regularity theory. We describe a series of recent results on the regularity theory of the Isaacs equation. These cover estimates in Holder and Sobolev spaces. We argue through a genuinely geometrical method, by importing information from a related Bellman equation.

In this thesis we study the optimal stochastic control problem of the drift of a Lévy process. We show that, for a broad class of Lévy processes, the partial integro-differential Hamilton-Jacobi-Bellman equation for the value function admits classical solutions and that control policies exist in feedback form. We then explore the class of Lévy processes that satisfy the requirements of the theorem, and find connections between the uniform integrability requirement and the notions of the score function and Fisher information from information theory. Finally we present three different numerical implementations of the control problem: a traditional dynamic programming approach, and two iterative approaches, one based on a finite difference scheme and the other on the Fourier transform. / text

Cette thèse est consacrée à la résolution des problèmes de contrôle non linéaires par des méthodes de calcul hybride. L'idée défendue est que la modélisation par les systèmes hybrides permet la résolution approchée des problèmes non linéaires sans connaissance a priori du comportement du système étudié. Dans un premier temps, nous nous intéressons à la modélisation des systèmes de contrôle non linéaires par une nouvelle classe de systèmes hybrides affines par morceaux. Un soin particulier est apporté à l'étude de l'erreur et de la convergence de l'approximation hybride. La deuxième partie est consacrée au problème de la contrôlabilité à l'origine des systèmes non linéaires. Nous nous intéressons tout d'abord à la quantification de l'erreur commise entre le domaine contrôlable non linéaire et son approximation hybride. Nous proposons ensuite une approche constructive pour le calcul du domaine contrôlable, permettant alors de réduire l'exploration des états discrets de l'automate hybride. La dernière partie est dédiée à la recherche de solutions optimales des problèmes de contrôle non linéaires et hybrides. Nous justifions tout d'abord la pertinence de la modélisation hybride à travers deux approches : le principe du maximum de Pontryagin et les solutions de viscosité des équations d'Hamilton-Jacobi-Bellman. Nous énonçons en particulier un principe du maximum hybride qui nous permet alors de déterminer la structure du contrôle optimal hybride. Ces trois parties répondent à un objectif principal : développer par le calcul hybride combinant analyse numérique et calcul formel, des outils mathématiques et algorithmiques efficaces pour l'étude de dynamiques contrôlées non linéaires.

This Master thesis was written in collaboration with the Kalmar-based consulting company Sederkvist Kommunikation. The purpose was to explore the market potential for Sederkvist Kommunikations new software Bellman Marketing. Based on these results a launch plan was created and advoice were given to Sederkvist Kommunikation as to how their new administrative tool could be introduced on the market. Data were collected through semi-structured telephone interviews. The population was Swedish so called Gazelle-companies (growth-companies listed yearly by the business newspaper Dagens Industri) and totally 27 units were examined. Analysis indicated that there was a relatively large market potential for Bellman Marketing among Swedish fast-growing companies. The market does however not perceive a need for the software at this point, thus this need has to be pointed out first by Sederkvist Kommunikation. By effective market segmentation and with messages suited for target customers, the launch of Bellman Marketing can be successful.

<p>This Master thesis was written in collaboration with the Kalmar-based consulting company Sederkvist Kommunikation. The purpose was to explore the market potential for Sederkvist Kommunikations new software Bellman Marketing. Based on these results a launch plan was created and advoice were given to Sederkvist Kommunikation as to how their new administrative tool could be introduced on the market. Data were collected through semi-structured telephone interviews. The population was Swedish so called Gazelle-companies (growth-companies listed yearly by the business newspaper Dagens Industri) and totally 27 units were examined. Analysis indicated that there was a relatively large market potential for Bellman Marketing among Swedish fast-growing companies. The market does however not perceive a need for the software at this point, thus this need has to be pointed out first by Sederkvist Kommunikation. By effective market segmentation and with messages suited for target customers, the launch of Bellman Marketing can be successful.</p>

>Magister Scientiae - MSc / Bank asset management mainly involves profit maximization through invest- ment in loans giving high returns on loans, investment in securities for reducing risk and providing liquidity needs. In particular, commercial banks grant loans to creditors who pay high interest rates and are not likely to default on their loans. Furthermore, the banks purchase securities with high returns and low risk. In addition, the banks attempt to lower risk by diversifying their asset portfolio. The main categories of assets held by banks are loans, treasuries (bonds issued by the national treasury), reserves and intangible assets. In this mini-thesis, we solve an optimal asset allocation problem in banking under the mean-variance frame work. The dynamics of the different assets are modelled as geometric Brownian motions, and our optimization problem is of the mean- variance type. We assume the Basel II regulations on banking supervision. In this contribution, the bank funds are invested into loans and treasuries with the main objective being to obtain an optimal return on the bank asset port- folio given a certain risk level. There are two main approaches to portfolio optimization, which are the so called martingale method and Hamilton Jacobi Bellman method. We shall follow the latter. As is common in portfolio op- timization problems, we obtain an explicit solution for the value function in the Hamilton Jacobi Bellman equation. Our approach to the portfolio prob- lem is similar to the presentation in the paper [Hojgaard, B., Vigna, E., 2007. Mean-variance portfolio selection and efficient frontier for defined contribution pension schemes. ISSN 1399-2503. On-line version ISSN 1601-7811]. We pro- vide much more detail and we make the application to banking. We illustrate our findings by way of numerical simulations.

Optimal control problems can in many cases become complicated and difficult to solve. One particular class of difficult control problems to solve are singular control problems. Standard methods for solving optimal control are discussed showing why those methods are difficult to apply to singular control problems. Then standard methods for solving singular control problems are discussed including why the standard methods can be difficult and often impossible to apply without having to resort to numerical techniques. Finally, an alternative method to solving a class of singular optimal control problems is given for a specific class of problems.

Asymptotic stability is a key notion of system stability for controlled dynamical systems as it guarantees that the system trajectories are bounded in a neighborhood of a given isolated equilibrium point and converge to this equilibrium over the infinite horizon. In some applications, however, asymptotic stability is not the appropriate notion of stability. For example, for systems with a continuum of equilibria, every neighborhood of an equilibrium contains another equilibrium and a nonisolated equilibrium cannot be asymptotically stable. Alternatively, in stabilization of spacecraft dynamics via gimballed gyroscopes, it is desirable to find state- and output-feedback control laws that guarantee partial-state stability of the closed-loop system, that is, stability with respect to part of the system state. Furthermore, we may additionally require finite-time stability of the closed-loop system, that is, convergence of the system's trajectories to a Lyapunov stable equilibrium in finite time. The Hamilton-Jacobi-Bellman optimal control framework provides necessary and sufficient conditions for the existence of state-feedback controllers that minimize a given performance measure and guarantee asymptotic stability of the closed-loop system. In this research, we provide extensions of the Hamilton-Jacobi-Bellman optimal control theory to develop state-feedback control laws that minimize nonlinear-nonquadratic performance criteria and guarantee semistability, partial-state stability, finite-time stability, and finite-time partial state stability of the closed-loop system.