Optimal multiple stopping time problem

We study the optimal multiple stopping time problem defined for each stopping time $S$ by $v(S)=\operatorname {ess}\sup_{\tau_1,...,\tau_d\geq S}E[\psi(\tau_1,...,\tau_d)|\mathcal{F}_S]$. The key point is the construction of a new reward $\phi$ such that the value function $v(S)$ also satisfies $v(S)=\operatorname {ess}\sup_{\theta\geq S}E[\phi(\theta)|\mathcal{F}_S]$. This new reward $\phi$ is not a right-continuous adapted process as in the classical case, but a family of random variables. For such a reward, we prove a new existence result for optimal stopping times under weaker assumptions than in the classical case. This result is used to prove the existence of optimal multiple stopping times for $v(S)$ by a constructive method. Moreover, under strong regularity assumptions on $\psi$, we show that the new reward $\phi$ can be aggregated by a progressive process. This leads to new applications, particularly in finance (applications to American options with multiple exercise times).Comment: Published in at http://dx.doi.org/10.1214/10-AAP727 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal double stopping time

We consider the optimal double stopping time problem defined for each stopping time $S$ by v(S)=\esssup\{E[\psi(\tau_1, \tau_2) | \F_S], \tau_1, \tau_2 \geq S \}. Following the optimal one stopping time problem, we study the existence of optimal stopping times and give a method to compute them. The key point is the construction of a {\em new reward} $\phi$ such that the value function $v(S)$ satisfies v(S)=\esssup\{E[\phi(\tau) | \F_S], \tau \geq S \}. Finally, we give an example of an american option with double exercise time.Comment: 6 page

A convergent scheme for a non local Hamilton-Jacobi equation modelling dislocation dynamic

We study dislocation dynamics with a level set point of view. The model we present here looks at the zero level set of the solution of a non local Hamilton Jacobi equation, as a dislocation in a plane of a crystal. The front has a normal speed, depending on the solution itself. We prove existence and uniqueness for short time in the set of continuous viscosity solutions. We also present a first order finite difference scheme for the corresponding level set formulation of the model. The scheme is based on monotone numerical Hamiltonian, proposed by Osher and Sethian. The non local character of the problem makesit not monotone. We obtain an explicit convergence rate of the approximate solution to the viscosity solution. We finally provide numerical simulations

La Cour de justice précise les conditions de bénéfice de la protection renforcée contre l’éloignement

L’arrêt rendu par la Grande Chambre de la CJUE le 17 avril 2018 laisse son lecteur perplexe. La Cour de justice était amenée à préciser les conditions de bénéfice de la protection renforcée contre l’éloignement prévue à l’article 28, paragraphe 3, de la directive 2004/38. Si la Cour estime que le droit de séjour permanent est un préalable indispensable à une telle protection, elle estime, toutefois, que l’emprisonnement ne prive pas automatiquement le citoyen du bénéfice de celle-ci

Optimal multiple stopping time problem

We study the optimal multiple stopping time problem defined for each stopping time $S$ by \displaystyle{ v(S)=\esssup_ {\tau_1,\cdots,\tau_d \geq S } E[\psi( \tau_1,\cdots,\tau_d)\, |\,\F_S]\,}.\\ The key point is the construction of a {\em new reward} $\phi$ such that the value function $v(S)$ satisfies also v(S)=\esssup_ {\theta \geq S } \,E[\phi(\theta)\, |\,\F_S]\,. This new reward $\phi$ is not a right continuous adapted process as in the classical case but a family of random variables. For such a reward, we prove a new existence result of optimal stopping times under weaker assumptions than in the classical case. This result is used to prove the existence of optimal multiple stopping times for $v(S)$ by a constructive method. Moreover, under strong regularity assumptions on $\psi$, we show that the new reward $\phi$ can be aggregated by a progressive process. This leads to different applications in particular in finance for American options with multiple exercise times

Preparation and optical properties of GA(x)IN(1-x)P alloys

The solution crystallization method was used to obtain Ga(z)In(1-x)P alloys for all values of x desired between zero and 1. The method of preparation makes it possible to crystallize the solid at a constant temperature. The points obtained by cathodoluminescence are nearly in straight lines. The optical absorption thresholds are confirmed in the results and make it possible to define the nature of the transitions except when x is in the neighborhood of 0.65. These determinations agree with those of Hilsum and Porteus (1968), but are not in agreement with those obtained by Lorenz et al, (1968)

Colchicum cupanii Guss. subsp. glossophyllum (Heldr.) Rouy, Datura innoxia Mill. and Eclipta prostrata (L.) L., new floristic records in Montenegro and western Balkan

During field investigations on Long Ulcinj Beach, Montenegrin coast, three taxa new to Montenegrin flora were collected: Colchicum cupanii Guss. subsp. glossophyllum (Heldr.) Rouy, Datura innoxia Mill. and Eclipta prostrata (L.) L. The first species is native to Europe, while the others are alien. Colchicum cupanii subsp. glossophyllum is an endemic of the Balkan Peninsula, with a distribution formerly restricted to Greece and Albania, so this new record extends its distribution to the north-west. The distribution of Eclipta prostrata in the Balkans is also enlarged by this new record for Montenegro, in a westerly direction

Plantes Nouvelles Pour La Flore Européenne

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Interfaces stationnaires pour les équations de Navier-Stokes

Nous présentons une étude complète des équations de Navier-Stokes stationnaire- s avec conditions aux limites de surface libre. Nous donnons des résultats d'existence, d'unicité et de régularité pour des écoulements de liquides visqueux à surface libre. Les techniques utilisées sont décrites en détail