>Magister Scientiae - MSc / The problem of optimal investment has been extensively studied by numerous researchers in order to generalize the original framework. Those generalizations have been made in different directions and using different techniques. For example, Perera [Optimal consumption, investment and insurance with insurable risk for an investor in a Levy market, Insurance: Mathematics and Economics, 46 (3) (2010) 479-484] applied the martingale approach to obtain a closed form solution for the optimal investment, consumption and insurance strategies of an individual in the presence of an insurable risk when the insurable risk and risky asset returns are described by Levy processes and the utility is a constant absolute risk aversion. In another work, Sattinger [The Markov consumption problem, Journal of Mathematical Economics, 47 (4-5) (2011) 409-416] gave a model of consumption behavior under uncertainty as the solution to a continuous-time dynamic control problem in which an individual moves between employment and unemployment according to a Markov process. In this thesis, we will review the consumption models in the above framework and will simulate some of them using an infinite series expansion method − a key focus of this research. Several numerical results obtained by using MATLAB are presented with detailed explanations.

Main goal of this thesis is improvement of an order book model so that it behaved more realistically, based on a model developed by J. Plačková in her diploma thesis in 2011. We consider this simple model for evolution of order book in which limit orders of unit size arrive according to independent Poisson processes. Frequency of buy limit orders below resp. sell limit orders above a given price level is described by demand and supply functions. Buy (resp. sell) limit orders that arrive with price above (resp. below) the current ask (resp. bid) price are converted into market orders and cancellation of orders is not allowed. We extend this model by introducing market makers who place at the same time one buy and one sell limit order with current bid and ask prices. We show how introducing market makers reduces the spread that in the original model was unrealistically large and also show a method of calculating the precise rate of market makers needed to reduce the spread to zero. 1

In recent times, credit risk analysis has grown to become one of the most important problems dealt with in the mathematical finance literature. Fundamentally, the problem deals with estimating the probability that an obligor defaults on their debt in a certain time. To obtain such a probability, several methods have been developed which are regulated by the Basel Accord. This establishes a legal framework for dealing with credit and market risks, and empowers banks to perform their own methodologies according to their interests under certain criteria. Credit risk analysis is founded on the rating system, which is an assessment of the capability of an obligor to make its payments in full and on time, in order to estimate risks and make the investor decisions easier.Credit risk models can be classified into several different categories. In structural form models (SFM), that are founded on the Black & Scholes theory for option pricing and the Merton model, it is assumed that default occurs if a firm's market value is lower than a threshold, most often its liabilities. The problem is that this is clearly is an unrealistic assumption. The factors models (FM) attempt to predict the random default time by assuming a hazard rate based on latent exogenous and endogenous variables. Reduced form models (RFM) mainly focus on the accuracy of the probability of default (PD), to such an extent that it is given more importance than an intuitive economical interpretation. Portfolio reduced form models (PRFM) belong to the RFM family, and were developed to overcome the SFM's difficulties.Most of these models are based on the assumption of having an underlying Markovian process, either in discrete or continuous time. For a discrete process, the main information is containted in a transition matrix, from which we obtain migration probabilities. However, according to previous analysis, it has been found that this approach contains embedding problems. The continuous time Markov process (CTMP) has its main information contained in a matrix Q of constant instantaneous transition rates between states. Both approaches assume that the future depends only on the present, though previous empirical analysis has proved that the probability of changing rating depends on the time a firm maintains the same rating. In order to face this difficulty we approach the PD with the continuous time semi-Markov process (CTSMP), which relaxes the exponential waiting time distribution assumption of the Markovian analogue.In this work we have relaxed the constant transition rate assumption and assumed that it depends on the residence time, thus we have derived CTSMP forward integral and differential equations respectively and the corresponding equations for the particular cases of exponential, gamma and power law waiting time distributions, we have also obtained a numerical solution of the migration probability by the Monte Carlo Method and compared the results with the Markovian models in discrete and continuous time respectively, and the discrete time semi-Markov process. We have focused on firms from U.S.A. and Canada classified as financial sector according to Global Industry Classification Standard and we have concluded that the gamma and Weibull distribution are the best adjustment models.

This research investigates a batch arrival queueing system with a Bernoulli scheduled vacation and random system breakdowns. It is assumed that the repair process does not start immediately after the breakdown. Consequently there maybe a delay in starting repairs. After every service completion the server may go on an optional vacation. When the original vacation is completed the server has the option to go on an extended vacation. It is assumed that the system is equipped with a stand-by server to serve the customers during the vacation period of the main server as well as during the repair process. The service times, vacation times, repair times, delay times and extended vacation times are assumed to follow different general distributions while the breakdown times and the service times of the stand-by server follow an exponential distribution. By introducing a supplementary variable we are able to obtain steady state results in an explicit closed form in terms of the probability generating functions. Some important performance measures including; the average length of the queue, the average number of customers in the system, the mean response time, and the value of the traffic intensity are presented. The professional MathCad 2001 software has been used to illustrate the numerical results in this study.

In this thesis, a model based on the continuous-time Markov process is built and implemented and later applied on an one chosen collective game. An extensive analysis of available data is carried out to build a regression model to estimate parameters of the game model. An usableness of the game model is shown by a simulation process. Pros and cons are evaluated in a comparison analysis against the application of the discrete-time Markov chains, how it was described in my bachelor thesis [Roman Zákutný (2007)]. In conclusion are discussed possible extensions for other collective games.

Neste trabalho procuramos determinar o controle ótimo para problemas de custo médio a longo prazo (CMLP) de sistemas lineares com saltos markovianos (SLSMs) com observação parcial dos estados da cadeia de Markov, e, para isso, implementamos métodos computacionais heurísticos como algoritmos evolutivos de primeira geração - algoritmo genético (AG) básico - e os algoritmos UMDA(Univariate Marginal Distribution Algorithm) e BOA(Bayesian Optimization Algorithm), de segunda geração. Utilizamos um algoritmo variacional para comparar com os métodos implementados e medir a qualidade de suas soluções. Desenvolvemos uma abordagem de transição de níveis de observação (ATNO), partindo de um problema de observação completa e migrando através de problemas parcialmente observados. Cada um dos métodos mencionados acima foi implementado também no contexto da ATNO. Para realizar uma análise estatística sobre o desempenho dos métodos computacionais, utilizamos um gerador de SLSMs com importantes características da teoria de controle como: estabilidade, estabilizabilidade, observabilidade, controlabilidade e detetabilidade. Por fim, apresentamos alguns resultados sobre o CMLP com controles estabilizantes e resultados parciais a respeito da unicidade de solução / In this work we are interested in the optimal control for the long run average cost (LRAC) problem for linear systems with Markov jump parameters (LSMJP), using heuristic methods like first generation evolutionary algorithms - genetic algorithm (GA) - and second generation algorithms including UMDA (Univariate Marginal Distribution Algorithm) and BOA (Bayesian Optimization Algorithm). We have developed a scheme that employs different problems with intermediate levels of observation of the Markov chain, starting with complete observation and shifting to the partial observation problem. The aforementioned methods have been implemented using this scheme. Moreover, in order to compare the methods, we use an algorithm for generating a number of LSMJP and we present a basic statistical analysis of the results. Finally, we present some results on the LRAC with stabilizing control and some partial results on the uniqueness of the solution

The need for high computing resources is on the rise, despite the exponential increase of the computing capacity of workstations, the proliferation of mobile devices, and the omnipresence of data centers with massive server farms that housed tens (if not hundreds) of thousands of powerful servers. This is mainly due to the unprecedented increase in the number of Internet users worldwide and the Internet of Things (IoTs). So far, Cloud Computing has been providing the necessary computing infrastructures for applications, including IoT applications. However, the current cloud infrastructures that are based on dedicated datacenters are expensive to set-up; running the infrastructure needs expertise, a lot of electrical power for cooling the facilities, and redundant supply of everything in a data center to provide the desired resilience. Moreover, the current centralized cloud infrastructures will not suffice for IoT's network intensive applications with very fast response requirements. Alternative cloud computing models that depend on spare resources of volunteer computers are emerging, including volunteer cloud computing, in addition to the conventional data center based clouds. These alternative cloud models have one characteristic in common -- they do not rely on dedicated data centers to provide the cloud services. Volunteer clouds are opportunistic cloud systems that run over donated spare resources of volunteer computers. On the one hand, volunteer clouds claim numerous outstanding advantages: affordability, on-premise, self-provision, greener computing (owing to consolidate use of existent computers), etc. On the other hand, full-fledged implementation of volunteer cloud computing raises unique technical and research challenges: management of highly dynamic and heterogeneous compute resources, Quality of Service (QoS) assurance, meeting Service Level Agreement (SLA), reliability, security/trust, which are all made more difficult due to the high dynamics and heterogeneity of the non-dedicated cloud hosts. This dissertation investigates the resource management aspect of volunteer cloud computing. Due to the intermittent availability and heterogeneity of computing resource involved, resource management is one of the challenging tasks in volunteer cloud computing. The dissertation, specifically, focuses on the Resource Discovery and VM Placement tasks of resource management. The resource base over which volunteer cloud computing depends on is a scavenged, sporadically available, aggregate computing power of individual volunteer computers. Delivering reliable cloud services over these unreliable nodes is a big challenge in volunteer cloud computing. The fault tolerance of the whole system rests on the reliability and availability of the infrastructure base. This dissertation discusses the modelling of a fault tolerant prediction based resource discovery in volunteer cloud computing. It presents a multi-state semi-Markov process based model to predict the future availability and reliability of nodes in volunteer cloud systems. A volunteer node is modelled as a semi-Markov process, whose future state depends only on its current state. This exactly matches with a key observation made in analyzing the traces of personal computers in enterprises that the daily patterns of resource availability are comparable to those in the most recent days. The dissertation illustrates how prediction based resource discovery enables volunteer cloud systems to provide reliable cloud services over the unreliable and non-dedicated volunteer hosts with empirical evidences. VM placement algorithms play crucial role in Cloud Computing in fulfilling its characteristics and achieving its objectives. In general, VM placement is a challenging problem that has been extensively studied in conventional Cloud Computing context. Due to its divergent characteristics, volunteer cloud computing needs a novel and unique way of solving the existing Cloud Computing problems, including VM placement. Intermittent availability of nodes, unreliable infrastructure, and resource constrained nodes are some of the characteristics of volunteer cloud computing that make VM placement problem more complicated. In this dissertation, we model the VM placement problem as a \textit{Bounded 0-1 Multi-Dimensional Knapsack Problem}. As a known NP-hard problem, the dissertation discusses heuristic based algorithms that takes the typical characteristics of volunteer cloud computing into consideration, to solve the VM placement problem formulated as a knapsack problem. Three algorithms are developed to meet the objectives and constraints specific to volunteer cloud computing. The algorithms are tested on a real volunteer cloud computing test-bed and showed a good performance results based on their optimization objectives. The dissertation also presents the design and implementation of a real volunteer cloud computing system, cuCloud, that bases its resource infrastructure on donated computing resource of computers. The need for the development of cuCloud stems from the lack of experimentation platform, real or simulation, that specifically works for volunteer cloud computing. The cuCloud is a system that can be called a genuine volunteer cloud computing system, which manifests the concept of ``Volunteer Computing as a Service'' (VCaaS), with a particular significance in edge computing and related applications. In the course of this dissertation, empirical evaluations show that volunteer clouds can be used to execute range of applications reliably and efficiently. Moreover, the physical proximity of volunteer nodes to where applications originate, edge of the network, helps them in reducing the round trip time latency of applications. However, the overall computing capability of volunteer clouds will not suffice to handle highly resource intensive applications by itself. Based on these observations, the dissertation also proposes the use of volunteer clouds as a resource fabric in the emerging Edge Computing paradigm as a future work.

L’objectif de cette thèse est de proposer une méthode permettant d’optimiser la stratégie de maintenance d’un système multi-composants. Cette nouvelle stratégie doit être adaptée aux conditions d’utilisation et aux contraintes budgétaires et sécuritaires. Le vieillissement des composants et la complexité des stratégies de maintenance étudiées nous obligent à avoir recours à de nouveaux modèles probabilistes afin de répondre à la problématique. Nous utilisons un processus stochastique issu de la Fiabilité Dynamique nommé processus markovien déterministe par morceaux (Piecewise Deterministic Markov Process ou PDMP). L’évaluation des quantités d’intérêt (fiabilité, nombre moyen de pannes...) est ici réalisé à l’aide d’un algorithme déterministe de type volumes finis. L’utilisation de ce type d’algorithme, dans ce cadre d’application, présente des difficultés informatiques dues à la place mémoire. Nous proposons plusieurs méthodes pour repousser ces difficultés. L’optimisation d’un plan de maintenance est ensuite effectuée à l’aide d’un algorithme de recuit simulé. Cette méthodologie a été adaptée à deux systèmes ferroviaires utilisés par la SNCF, l’un issu de l’infrastructure, l’autre du matériel roulant. / The aim of this work is to propose a methodology to optimize a multi-components system maintenance. This new maintenance strategy must be adapted to budget and safety constraints and operating conditions. The aging of components and the complexity of studied maintenance strategies require us to use new probabilistic models in order to address the problem. A stochastic process from Dynamic Reliability calculations are here established by using a deterministic algorithm method based on a finite volume scheme. Using this type of algorithm in this context of application presents difficulties due to computer memory space. We propose several methods to counter these difficulties. The optimization of a maintenance plan is then performed using simulated annealing algorithm. This methodology was used to optimize the maintenance of two rail systems used by the French national railway company (SNCF).

This thesis consists of three parts. In the first part, we recall some background theory that will be used throughout the thesis. In the second part, we studied the existence and uniqueness of solutions of the stochastic heat equations with Markovian switching. In the third part, we investigate the properties of solutions, such as Feller property, strong Feller property and stability.

In the presented work we study the existence of periodic solution to infinite dimensional stochastic equation with periodic coefficients driven by Cylindrical Wiener process. Used theory of infinite dimensional stochastic equations in Hilbert spaces and Markov processes is summarized in the first two chapters. In the third and last chapter we present the result itself. Necessary technical background mostly from operator theory is encapsulated in the Appendix. The proof of existence of periodic solution of corresponding equation is a combination of arguments by Khasminskii, which ensure under suitable conditions the existence of periodic Markov process, and the results of Da Prato, G¸atatrek and Zabczyk for the existence of invariant measure for homogeneous stochastic equation in Hilbert spaces. At the end we derive sufficient condition for the existence of periodic solution in the language of coefficients using the work of Ichikawa and illustrate the results by the example of Stochastic PDE. The work is written in English.