On the future infimum of positive self-similar Markov processes.

We establish integral tests and laws of the iterated logartihm for the upper envelope of the future infimum of positive self-similar Markov processes and for increasing self-similar Markov processes at $0$ an at $+\infty$. our proofs are based on the Lamperti transformation and time reversal arguments due to Chaumont and Pardo [9]. These results extend laws of the iterated logarithm for the future infimum of a Bessel process due to Khoshnevisan et al. [11]

The upper envelope of positive self-similar Markov processes

We establish integral tests and laws of the iterated logarithm at 0 and at $+\infty$, for the upper envelope of positive self-similar Markov processes. Our arguments are based on the Lamperti representation, time reversal arguments and on the study of the upper envelope of their future infimum due to Pardo \cite{Pa}. These results extend integral test and laws of the iterated logarithm for Bessel processes due to Dvoretsky and Erd\"os \cite{de} and stable L\'evy processes conditioned to stay positive with no positive jumps due to Bertoin \cite{be1}

Some explicit identities associated with positive self-similar Markov processes

We consider some special classes of L\'evy processes with no gaussian component whose L\'evy measure is of the type $\pi(dx)=e^{\gamma x}\nu(e^x-1) dx$, where $\nu$ is the density of the stable L\'evy measure and $\gamma$ is a positive parameter which depends on its characteristics. These processes were introduced in \cite{CC} as the underlying L\'evy processes in the Lamperti representation of conditioned stable L\'evy processes. In this paper, we compute explicitly the law of these L\'evy processes at their first exit time from a finite or semi-finite interval, the law of their exponential functional and the first hitting time probability of a pair of points

On the extinction of Continuous State Branching Processes with catastrophes

We consider continuous state branching processes (CSBP) with additional multiplicative jumps modeling dramatic events in a random environment. These jumps are described by a L\'evy process with bounded variation paths. We construct a process of this class as the unique solution of a stochastic differential equation. The quenched branching property of the process allows us to derive quenched and annealed results and to observe new asymptotic behaviors. We characterize the Laplace exponent of the process as the solution of a backward ordinary differential equation and establish the probability of extinction. Restricting our attention to the critical and subcritical cases, we show that four regimes arise for the speed of extinction, as in the case of branching processes in random environment in discrete time and space. The proofs are based on the precise asymptotic behavior of exponential functionals of L\'evy processes. Finally, we apply these results to a cell infection model and determine the mean speed of propagation of the infection

Extinction and coming down from infinity of CB-processes with competition in a Lévy environment

In this note, we are interested on the event of extinction and the property of coming down from infinity of continuous state branching (or CB for short) processes with competition in a Lévy environment whose branching mechanism satisfies the so-called Grey's condition. In particular, we deduce, under the assumption that the Lévy environment does not drift towards infinity, that for any starting point the process becomes extinct in finite time a.s. Moreover if we impose an integrability condition on the competition mechanism, then the process comes down from infinity regardless the long term behaviour of the environment

The extended hypergeometric class of Lévy processes

We review and extend the class of hypergeometric Lévy processes explored in Kuznetsov and Pardo (2013) with a view to computing fluctuation identities related to stable processes. We give the Wiener-Hopf factorisation of a process in the extended class, characterise its exponential functional, and give three concrete examples arising from transformations of stable processes. <br/

Fluctuations for matrix-valued Gaussian processes

We consider a symmetric matrix-valued Gaussian process $Y^{(n)}=(Y^{(n)}(t);t\ge0)$ and its empirical spectral measure process $\mu^{(n)}=(\mu_{t}^{(n)};t\ge0)$. Under some mild conditions on the covariance function of $Y^{(n)}$, we find an explicit expression for the limit distribution of $$Z_F^{(n)} := \left( \big(Z_{f_1}^{(n)}(t),\ldots,Z_{f_r}^{(n)}(t)\big) ; t\ge0\right),$$ where $F=(f_1,\dots, f_r)$, for $r\ge 1$, with each component belonging to a large class of test functions, and Z_{f}^{(n)}(t) := n\int_{\R}f(x)\mu_{t}^{(n)}( x)-n\E\left[\int_{\R}f(x)\mu_{t}^{(n)}( x)\right]. More precisely, we establish the stable convergence of $Z_F^{(n)}$ and determine its limiting distribution. An upper bound for the total variation distance of the law of $Z_{f}^{(n)}(t)$ to its limiting distribution, for a test function $f$ and $t\geq0$ fixed, is also given

Juxtaposing BTE and ATE – on the role of the European insurance industry in funding civil litigation

One of the ways in which legal services are financed, and indeed shaped, is through private insurance arrangement. Two contrasting types of legal expenses insurance contracts (LEI) seem to dominate in Europe: before the event (BTE) and after the event (ATE) legal expenses insurance. Notwithstanding institutional differences between different legal systems, BTE and ATE insurance arrangements may be instrumental if government policy is geared towards strengthening a market-oriented system of financing access to justice for individuals and business. At the same time, emphasizing the role of a private industry as a keeper of the gates to justice raises issues of accountability and transparency, not readily reconcilable with demands of competition. Moreover, multiple actors (clients, lawyers, courts, insurers) are involved, causing behavioural dynamics which are not easily predicted or influenced. Against this background, this paper looks into BTE and ATE arrangements by analysing the particularities of BTE and ATE arrangements currently available in some European jurisdictions and by painting a picture of their respective markets and legal contexts. This allows for some reflection on the performance of BTE and ATE providers as both financiers and keepers. Two issues emerge from the analysis that are worthy of some further reflection. Firstly, there is the problematic long-term sustainability of some ATE products. Secondly, the challenges faced by policymakers that would like to nudge consumers into voluntarily taking out BTE LEI

Differential cross section measurements for the production of a W boson in association with jets in proton–proton collisions at √s = 7 TeV

Measurements are reported of differential cross sections for the production of a W boson, which decays into a muon and a neutrino, in association with jets, as a function of several variables, including the transverse momenta (pT) and pseudorapidities of the four leading jets, the scalar sum of jet transverse momenta (HT), and the difference in azimuthal angle between the directions of each jet and the muon. The data sample of pp collisions at a centre-of-mass energy of 7 TeV was collected with the CMS detector at the LHC and corresponds to an integrated luminosity of 5.0 fb[superscript −1]. The measured cross sections are compared to predictions from Monte Carlo generators, MadGraph + pythia and sherpa, and to next-to-leading-order calculations from BlackHat + sherpa. The differential cross sections are found to be in agreement with the predictions, apart from the pT distributions of the leading jets at high pT values, the distributions of the HT at high-HT and low jet multiplicity, and the distribution of the difference in azimuthal angle between the leading jet and the muon at low values.United States. Dept. of EnergyNational Science Foundation (U.S.)Alfred P. Sloan Foundatio