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

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 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

Finite-Element Discretization of Static Hamilton-Jacobi Equations Based on a Local Variational Principle

We propose a linear finite-element discretization of Dirichlet problems for static Hamilton-Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local variational principle. It generalizes several approaches known in the literature and allows for a simple and transparent convergence theory. In this paper the resulting system of nonlinear equations is solved by an adaptive Gauss-Seidel iteration that is easily implemented and quite effective as a couple of numerical experiments show.Comment: 19 page

Complete Genome Sequence of the Piezophilic, Mesophilic, Sulfate-Reducing Bacterium Desulfovibrio hydrothermalis AM13(T.).

International audienceDesulfovibrio hydrothermalis AM13(T) is a piezophilic, mesophilic, hydrogenotrophic sulfate-reducing bacterium collected from a deep-sea hydrothermal chimney on the East Pacific Rise (2,600 m depth, 13°N). We report the genome sequence of this bacterium, which includes a 3,702,934-bp chromosome and a circular plasmid of 5,328 bp

Transcriptome analysis of Thapsia laciniata rouy provides insights into terpenoid biosynthesis and diversity in apiaceae

Thapsia laciniata Rouy (Apiaceae) produces irregular and regular sesquiterpenoids with thapsane and guaiene carbon skeletons, as found in other Apiaceae species. A transcriptomic analysis utilizing Illumina next-generation sequencing enabled the identification of novel genes involved in the biosynthesis of terpenoids in Thapsia. From 66.78 million HQ paired-end reads obtained from T. laciniata roots, 64.58 million were assembled into 76,565 contigs (N50: 1261 bp). Seventeen contigs were annotated as terpene synthases and five of these were predicted to be sesquiterpene synthases. Of the 67 contigs annotated as cytochromes P450, 18 of these are part of the CYP71 clade that primarily performs hydroxylations of specialized metabolites. Three contigs annotated as aldehyde dehydrogenases grouped phylogenetically with the characterized ALDH1 from Artemisia annua and three contigs annotated as alcohol dehydrogenases grouped with the recently described ADH1 from A. annua. ALDH1 and ADH1 were characterized as part of the artemisinin biosynthesis. We have produced a comprehensive EST dataset for T. laciniata roots, which contains a large sample of the T. laciniata transcriptome. These transcriptome data provide the foundation for future research into the molecular basis for terpenoid biosynthesis in Thapsia and on the evolution of terpenoids in Apiaceae.Damian Paul Drew, Bjørn Dueholm, Corinna Weitzel, Ye Zhang, Christoph W. Sensen and Henrik Toft Simonse

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

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