The SuperB factory: Physics Prospects and Project Status

I will briefly review of some highlights of the SuperB physics programme, the status of the accelerator and detector studies, and the future plans.Comment: 6 pages, 0 figures, contribution to the Physics in Collision 2012 (PIC2012) conference. arXiv admin note: substantial text overlap with arXiv:1112.1394, arXiv:1012.244

On the optimality of Periodic barrier strategies for a spectrally positive L\'evy process

We study the optimal dividend problem in the dual model where dividend payments can only be made at the jump times of an independent Poisson process. In this context, Avanzi et al. [5] solved the case with i.i.d. hyperexponential jumps; they showed the optimality of a (periodic) barrier strategy where dividends are paid at dividend-decision times if and only if the surplus is above some level. In this paper, we generalize the results for a general spectrally positive Levy process with additional terminal payoff/penalty at ruin, and also solve the case with classical bail-outs so that the surplus is restricted to be nonnegative. The optimal strategies as well as the value functions are concisely written in terms of the scale function. Numerical results are also given.Comment: To appear on Insurance: Mathematics and Economic

FF-jumping and FF-Jacobian ideals for hypersurfaces

We introduce two families of ideals, $F$-jumping ideals and $F$-Jacobian ideals, in order to study the singularities of hypersurfaces in positive characteristic. Both families are defined using the $D$-modules $M_{\alpha}$ that were introduced by Blickle, Musta\c{t}\u{a} and Smith. Using strong connections between $F$-jumping ideals and generalized test ideals, we give a characterization of $F$-jumping numbers for hypersurfaces. Furthermore, we give an algorithm that determines whether certain numbers are $F$-jumping numbers. In addition, we use $F$-Jacobian ideals to study intrinsic properties of the singularities of hypersurfaces. In particular, we give conditions for $F$-regularity. Moreover, $F$-Jacobian ideals behave similarly to Jacobian ideals of polynomials. Using techniques developed to study these two new families of ideals, we provide relations among test ideals, generalized test ideals, and generalized Lyubeznik numbers for hypersurfaces.Comment: References updated, 32 page

Refraction-reflection strategies in the dual model

We study the dual model with capital injection under the additional condition that the dividend strategy is absolutely continuous. We consider a refraction-reflection strategy that pays dividends at the maximal rate whenever the surplus is above a certain threshold, while capital is injected so that it stays positive. The resulting controlled surplus process becomes the spectrally positive version of the refracted-reflected process recently studied by P\'erez and Yamazaki (2015). We study various fluctuation identities of this process and prove the optimality of the refraction-reflection strategy. Numerical results on the optimal dividend problem are also given.Comment: 33 page

On the Refracted-Reflected Spectrally Negative L\'evy Processes

We study a combination of the refracted and reflected L\'evy processes. Given a spectrally negative L\'evy process and two boundaries, it is reflected at the lower boundary while, whenever it is above the upper boundary, a linear drift at a constant rate is subtracted from the increments of the process. Using the scale functions, we compute the resolvent measure, the Laplace transform of the occupation times as well as other fluctuation identities that will be useful in applied probability including insurance, queues, and inventory management.Comment: 28 pages, forthcoming in Stochastic Processes and their Application

Universality classes for general random matrix flows

We consider matrix-valued processes described as solutions to stochastic differential equations of very general form. We study the family of the empirical measure-valued processes constructed from the corresponding eigenvalues. We show that the family indexed by the size of the matrix is tight under very mild assumptions on the coefficients of the initial SDE. We characterize the limiting distributions of its subsequences as solutions to an integral equation. We use this result to study some universality classes of random matrix flows. These generalize the classical results related to Dyson Brownian motion and squared Bessel particle systems. We study some new phenomenons as the existence of the generalized Marchenko-Pastur distributions supported on the real line. We also introduce universality classes related to generalized geometric matrix Brownian motions and Jacobi processes. Finally we study, under some conditions, the convergence of the empirical measure-valued process of eigenvalues associated to matrix flows to the law of a free diffusion.Comment: 27 page

On the Free Fractional Wishart Process

We investigate the process of eigenvalues of a fractional Wishart process defined as N=B*B, where B is a matrix fractional Brownian motion recently studied by Nualart and P\'erez-Abreu. Using stochastic calculus with respect to the Young integral we show that the eigenvalues do not collide at any time with probability one. When the matrix process B has entries given by independent fractional Brownian motions with Hurst parameter $H\in(1/2,1)$ we derive a stochastic differential equation in a Malliavin calculus sense for the eigenvalues of the corresponding fractional Wishart process. Finally a functional limit theorem for the empirical measure-valued process of eigenvalues of a fractional Wishart process is obtained. The limit is characterized and referred to as the free fractional Wishart process which constitutes the family of fractional dilations of the free Poisson distribution

American options under periodic exercise opportunities

In this paper, we study a version of the perpetual American call/put option where exercise opportunities arrive only periodically. Focusing on the exponential L\'evy models with i.i.d. exponentially-distributed exercise intervals, we show the optimality of a barrier strategy that exercises at the first exercise opportunity at which the asset price is above/below a given barrier. Explicit solutions are obtained for the cases the underlying L\'evy process has only one-sided jumps

A Random Matrix Approximation for the Non-commutative Fractional Brownian Motion

A functional limit theorem for the empirical measure-valued process of eigenvalues of a matrix fractional Brownian motion is obtained. It is shown that the limiting measure-valued process is the non-commutative fractional Brownian motion recently introduced by Nourdin and Taqqu. Young and Skorohod stochastic integral techniques and fractional calculus are the main tools used

On the intersection graph of the disks with diameters the sides of a convex nn-gon

Given a convex $n$-gon, we can draw $n$ disks (called side disks) where each disk has a different side of the polygon as diameter and the midpoint of the side as its center. The intersection graph of such disks is the undirected graph with vertices the $n$ disks and two disks are adjacent if and only if they have a point in common. Such a graph was introduced by Huemer and P\'erez-Lantero in 2016, proved to be planar and Hamiltonian. In this paper we study further combinatorial properties of this graph. We prove that the treewidth is at most 3, by showing an $O(n)$-time algorithm that builds a tree decomposition of width at most 3, given the polygon as input. This implies that we can construct the intersection graph of the side disks in $O(n)$ time. We further study the independence number of this graph, which is the maximum number of pairwise disjoint disks. The planarity condition implies that for every convex $n$-gon we can select at least $\lceil n/4 \rceil$ pairwise disjoint disks, and we prove that for every $n\ge 3$ there exist convex $n$-gons in which we cannot select more than this number. Finally, we show that our class of graphs includes all outerplanar Hamiltonian graphs except the cycle of length four, and that it is a proper subclass of the planar Hamiltonian graphs