Optimal stopping for Lévy processes and affine functions

32 pagesOptimal stoppingThis paper studies an optimal stopping problem for Lévy processes. We give a justification of the form of the Snell envelope using standard results of optimal stopping. We also justify the convexity of the value function, and without a priori restriction to a particular class of stopping times, we deduce that the smallest optimal stopping time is necessarily a hitting time. We propose a method which allows to obtain the optimal threshold. Moreover this method allows to avoid long calculations of the integro-differential operatorused in the usual proofs

Arr\^et optimal pour les processus de Markov forts et les fonctions affines

In this Note we study optimal stopping problems for strong Markov processes and affine functions. We give a justification of the Snell envelope form using standard results of optimal stopping. We also justify the convexity of the value function, and without a priori restriction to a particular class of stopping times, we deduce that the smallest optimal stopping time is necessarily a hitting time

Optimal stopping for L\'evy processes and affine functions

This paper studies an optimal stopping problem for L\'evy processes. We give a justification of the form of the Snell envelope using standard results of optimal stopping. We also justify the convexity of the value function, and without a priori restriction to a particular class of stopping times, we deduce that the smallest optimal stopping time is necessarily a hitting time. We propose a method which allows to obtain the optimal threshold. Moreover this method allows to avoid long calculations of the integro-differential operatorused in the usual proofs.Comment: 32 page

A note on the computation of an actuarial Waring formula in the finite-exchangeable case

We present in this paper the actuarial Waring formula, which is used in several fields, like life-insurance or credit risk. In a particular framework where considered random variables are exchangeable, we show that some problems can occur when using this formula. We propose alternative recursions in order to improve the complexity of the calculations, and to cope with the numerical instability of the formula.

First passage time law for some jump-diffusion processes : existence of a density

Let (Xt, t >= 0) be a diffusion process with jumps, sum of a Brownian motion with drift and a compound Poisson process. We consider T_x the first hitting time of a fixed level x > 0 by (Xt, t >= 0). We prove that the law of T_x has a density (defective when E(X1) < 0) with respect to the Lebesgue measure

An extension of Davis and Lo's contagion model

International audienceThe present paper provides a multi-period contagion model in the credit risk field. Our model is an extension of Davis and Lo's infectious default model. We consider an economy of n firms which may default directly or may be infected by other defaulting firms (a domino effect being also possible). The spontaneous default without external influence and the infections are described by not necessarily independent Bernoulli-type random variables. Moreover, several contaminations could be required to infect another firm. In this paper we compute the probability distribution function of the total number of defaults in a dependency context. We also give a simple recursive algorithm to compute this distribution in an exchangeability context. Numerical applications illustrate the impact of exchangeability among direct defaults and among contaminations, on different indicators calculated from the law of the total number of defaults. We then examine the calibration of the model on iTraxx data before and during the crisis. The dynamic feature together with the contagion effect seem to have a significant impact on the model performance, especially during the recent distressed period

Modélisation du risque de défaut en entreprise

In the first part of this thesis, we study some optimal stopping time problems of the form : sup_{\tau\in \Delta, \tau\geq 0} \esp_v\left[g(V_{\tau})\right] \hbox{~or}~ sup_{\tau\in \Delta, \tau\geq 0} \esp_v\left[e^{-r\tau}\bar{g}(V_{\tau})\right], where $V$ is a stochastic process, $g$ and $\bar{g}$ two Borelian functions, $r>0$ and $\Delta$ is the set of \F^V_.-stopping times (\F^V being the filtration generated by the process $V$). These problems can be applied in Finance, Economy or Medicine.In the first part of this thesis we show that sometimes the smallest optimal stopping time is a hitting time. That's why, in the second part we study the hitting time law of a Lévy jump process. Some applications to finance are given : we compute the intensity of this stopping time associated with some filtration \F. Two cases are presented : when the stopping time is a \F-stopping time and when it is not.Dans une première partie, on étudie quelques problèmes d'arrêt optimal de la forme sup_{\tau\in \Delta, \tau\geq 0} \esp_v\left[g(V_{\tau})\right] \hbox{~ou}~ sup_{\tau\in \Delta, \tau\geq 0} \esp_v\left[e^{-r\tau}\bar{g}(V_{\tau})\right], où $V$ est un processus stochastique, $g$ et $\bar{g}$ deux fonctions boréliennes, $r>0$ et $\Delta$ est l'ensemble des \F^V-temps d'arrêt (\F_.^V étant la filtration engendrée par le processus $V$). L'étude de ces problèmes est motivée par les applications dans plusieurs domaines comme la finance, l'économie ou la médecine. La première partie est une mise en évidence du fait que le plus petit temps d'arrêt optimal est parfois un temps d'atteinte. C'est pourquoi, dans la deuxième partie de la thèse, on s'intéresse à la loi d'un temps d'atteinte d'un processus de Lévy à sauts ainsi qu'à quelques applications à la finance, plus précisément lors du calcul de l'intensité de ce temps d'arrêt associée à une certaine filtration \F. Deux cas sont présentés : quand le temps d'arrêt est un \F-temps d'arrêt et quand il ne l'est pas

The density of the ruin time for a renewal-reward process perturbed by a diffusion

International audienceLet $X$ be a mixed process, sum of a brownian motion and a renewal-reward process, and $\tau_{x}$ be the first passage time of a fixed level $x<0$ by $X$. We prove that $\tau_x$ has a density and we give a formula for it. Links with ruin theory are presented. Our result may be computed in classical settings (for a Lévy or Sparre Andersen process) and also in a non markovian context with possible positive and negative jumps. Some numerical applications illustrate the interest of this density formula

The density of the ruin time for a renewal-reward process perturbed by a diffusion

Let $X$ be a mixed process, sum of a brownian motion and a renewal-reward process, and $\tau_{x}$ be the first passage time of a fixed level $xRenewal-reward process ; Brownian motion ; Jump-diffusion process ; Time of ruin.