Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics

Exit problems for one-dimensional Lévy processes are easier when jumps only occur in one direction. In the last few years, this intuition became more precise: we know now that a wide variety of identities for exit problems of spectrally-negative Lévy processes may be ergonomically expressed in terms of two q-harmonic functions (or scale functions or positive martingales) W and Z. The proofs typically require not much more than the strong Markov property, which hold, in principle, for the wider class of spectrally-negative strong Markov processes. This has been established already in particular cases, such as random walks, Markov additive processes, Lévy processes with omega-state-dependent killing, and certain Lévy processes with state dependent drift, and seems to be true for general strong Markov processes, subject to technical conditions. However, computing the functions W and Z is still an open problem outside the Lévy and diffusion classes, even for the simplest risk models with state-dependent parameters (say, Ornstein–Uhlenbeck or Feller branching diffusion with phase-type jumps)

Zooming-in on a Lévy process: Failure to observe threshold exceedance over a dense grid

For a Lévy process X on a finite time interval consider the probability that it exceeds some fixed threshold x > 0 while staying below x at the points of a regular grid. We establish exact asymptotic behavior of this probability as the number of

Some Applications of Markov Additive Processes as Models in Insurance and Financial Mathematics

Cette thèse est principalement constituée de trois articles traitant des processus markoviens additifs, des processus de Lévy et d'applications en finance et en assurance. Le premier chapitre est une introduction aux processus markoviens additifs (PMA), et une présentation du problème de ruine et de notions fondamentales des mathématiques financières. Le deuxième chapitre est essentiellement l'article "Lévy Systems and the Time Value of Ruin for Markov Additive Processes" écrit en collaboration avec Manuel Morales et publié dans la revue European Actuarial Journal. Cet article étudie le problème de ruine pour un processus de risque markovien additif. Une identification de systèmes de Lévy est obtenue et utilisée pour donner une expression de l'espérance de la fonction de pénalité actualisée lorsque le PMA est un processus de Lévy avec changement de régimes. Celle-ci est une généralisation des résultats existant dans la littérature pour les processus de risque de Lévy et les processus de risque markoviens additifs avec sauts "phase-type". Le troisième chapitre contient l'article "On a Generalization of the Expected Discounted Penalty Function to Include Deficits at and Beyond Ruin" qui est soumis pour publication. Cet article présente une extension de l'espérance de la fonction de pénalité actualisée pour un processus subordinateur de risque perturbé par un mouvement brownien. Cette extension contient une série de fonctions escomptée éspérée des minima successives dus aux sauts du processus de risque après la ruine. Celle-ci a des applications importantes en gestion de risque et est utilisée pour déterminer la valeur espérée du capital d'injection actualisé. Finallement, le quatrième chapitre contient l'article "The Minimal entropy martingale measure (MEMM) for a Markov-modulated exponential Lévy model" écrit en collaboration avec Romuald Hervé Momeya et publié dans la revue Asia-Pacific Financial Market. Cet article présente de nouveaux résultats en lien avec le problème de l'incomplétude dans un marché financier où le processus de prix de l'actif risqué est décrit par un modèle exponentiel markovien additif. Ces résultats consistent à charactériser la mesure martingale satisfaisant le critère de l'entropie. Cette mesure est utilisée pour calculer le prix d'une option, ainsi que des portefeuilles de couverture dans un modèle exponentiel de Lévy avec changement de régimes.This thesis consists mainly of three papers concerned with Markov additive processes, Lévy processes and applications on finance and insurance. The first chapter is an introduction to Markov additive processes (MAP) and a presentation of the ruin problem and basic topics of Mathematical Finance. The second chapter contains the paper "Lévy Systems and the Time Value of Ruin for Markov Additive Processes" written with Manuel Morales and that is published in the European Actuarial Journal. This paper studies the ruin problem for a Markov additive risk process. An expression of the expected discounted penalty function is obtained via identification of the Lévy systems. The third chapter contains the paper "On a Generalization of the Expected Discounted Penalty Function to Include Deficits at and Beyond Ruin" that is submitted for publication. This paper presents an extension of the expected discounted penalty function in a setting involving aggregate claims modelled by a subordinator, and Brownian perturbation. This extension involves a sequence of expected discounted functions of successive minima reached by a jump of the risk process after ruin. It has important applications in risk management and in particular, it is used to compute the expected discounted value of capital injection. Finally, the fourth chapter contains the paper "The Minimal Entropy Martingale Measure (MEMM) for a Markov-Modulated Exponential" written with Romuald Hérvé Momeya and that is published in the journal Asia Pacific Financial Market. It presents new results related to the incompleteness problem in a financial market, where the risky asset is driven by Markov additive exponential model. These results characterize the martingale measure satisfying the entropy criterion. This measure is used to compute the price of the option and the portfolio of hedging in an exponential Markov-modulated Lévy model

Calcul de Malliavin, processus de Lévy et applications en finance : quelques contributions

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal

Convex hulls of Lévy processes in space time

In this thesis we apply a stick-breaking representation of the convex minorant and concave majorant of a one-dimensional Lévy process to show multiple probabilistic and geometric properties for the convex hull of a Lévy process. We show a central limit theorem for the fluctuations of the length of the concave majorant of a Lévy process when there is a finite second moment and consider the asymptotic dependence with the extrema of the process itself. The limit fluctuations of the length is also considered in the case where the Lévy process is in the domain of attraction of an α-stable law. In the rest of the thesis we study smoothness properties of the convex hull. Indeed, we characterise a class of Lévy processes whose graph has a continuously differentiable convex hull. Moreover, we also study how smooth the convex hull can be, by studying the growth rate of the convex minorant whenever the right derivative of the convex minorant increases continuously. Lastly, we characteris e the Hölder continuity of the convex hull of a one-dimensional Lévy process

Optimal prediction problems and the last zero of spectrally negative Lévy processes

In recent years the study of Levy processes has received considerable attention in the literature. In particular, spectrally negative Levy processes have applications in insurance, finance, reliability and risk theory. For instance, in risk theory, the capital of an insurance company over time is studied. A key quantity of interest is the moment of ruin, which is classically defined as the first passage time below zero. Consider instead the situation where after the moment of ruin the company may have funds to endure a negative capital for some time. In that case, the last time below zero becomes an important quantity to be studied. An important characteristic of last passage times is that they are random times which are not stopping times. This means that the information available at any time is not enough to determine its value and only with the whole realisation of the process that it can be determined. On the other hand, stopping times are random times such that its realisation can be derived only with the past information. Suppose that at any time period there is a need to know the value of a last passage time for some appropriate actions to be taken. It is then clear that an alternative to this problem is to approximate the last passage time with a stopping time such that they are close in some sense. In this work, we consider the optimal prediction to the last zero of a spectrally negative Levy process. This is equivalent to find a stopping time that minimises its distance with respect to the last time the process goes below zero. In order to fulfil this goal, we also study the last zero before at any fixed time and its dynamics as a process. Moreover, having in mind some applications in the insurance sector, we study the joint distribution of the number of downcrossings by jump and the local time before an exponential time

Stochastics of Environmental and Financial Economics

Systems Theory, Contro