This report is mainly based on the paper "Optimal multiple stopping and valuation of swing options" by R. Carmona and N. Touzi (1). Here the authors model and solve optimal stopping problems with more than one exercise time. The existence of optimal stopping times is firstly proved and they then construct the value function of American put options with multiple exercises in the case of the Black-Scholes model, characterizing the exercise boundaries of the perpetual case. Finally, they extend the analysis to the swing contracts with infinitely many exercise rights. In this report, we concentrate on explaining their rigorous mathematical analysis in detail, especially for the valuation of the perpetual American put options with single exercise and two exercise rights, and the characteristics of the exercise boundaries of the multiple stopping case. These results are presented as theorems in Chapter 2 and Chapter 3. / text

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.

The Autonomic Power System (APS) grand challenge was a multi-disciplinary EPSRC-funded research project that examined novel techniques that would enable the transition between today's and 2050's highly uncertain and complex energy network. Being part of the APS, this thesis reports on the sub-project 'RR2: Avoiding High-Impact Low Probability events'. The goal of RR2 is to develop new algorithms for controlling risk exposure to high-impact low probability (Hi-Lo) events through the provision of appropriate risk-sensitive control strategies. Additionally, RR2 is concerned with new techniques for identifying and modelling risk in future energy networks, in particular, the risk of Hi-Lo events. In this context, this thesis investigates two distinct problems arising from energy risk management. On the one hand, we examine the problem of finding managerial strategies for exercising the operational flexibility of energy assets. We look at this problem from a risk perspective taking into account non-linear risk preferences of energy asset managers. Our main contribution is the development of a risk-sensitive approach to the class of optimal switching problems. By recasting the problem as an iterative optimal stopping problem, we are able to characterise the optimal risk-sensitive switching strategies. As byproduct, we obtain a multiplicative dynamic programming equation for the value function, upon which we propose a numerical algorithm based on least squares Monte Carlo regression. On the other hand, we develop tools to identify and model the risk factors faced by energy asset managers. For this, we consider a class of models consisting of superposition of Gaussian and non-Gaussian Ornstein-Uhlenbeck processes. Our main contribution is the development of a Bayesian methodology based on Markov chain Monte Carlo (MCMC) algorithms to make inference into this class of models. On extensive simulations, we demonstrate the robustness and efficiency of the algorithms to different data features. Furthermore, we construct a diagnostic tool based on Bayesian p-values to check goodness-of-fit of the models on a Bayesian framework. We apply this tool to MCMC results from fitting historical electricity and gas spot price data- sets corresponding to the UK and German energy markets. Our analysis demonstrates that the MCMC-estimated models are able to capture not only long- and short-lived positive price spikes, but also short-lived negative price spikes which are typical of UK gas prices and German electricity prices. Combining together the solutions to the two problems above, we strive to capture the interplay between risk, uncertainty, flexibility and performance in various applications to energy systems. In these applications, which include power stations, energy storage and district energy systems, we consistently show that our risk management methodology offers a tradeoff between maximising average performance and minimising risk, while accounting for the jump dynamics of energy prices. Moreover, the tradeoff is achieved in such way that the benefits in terms of risk reduction outweigh the loss in average performance.