Given a spectrally negative Lévy process, we predict, in an $L_1$ sense, the last passage time of the process below zero before an independent exponential time. This optimal prediction problem generalises [2], where the infinite-horizon problem is solved. Using a similar argument as that in [24], we show that this optimal prediction problem is equivalent to solving an optimal prediction problem in a finite-horizon setting. Surprisingly (unlike the infinite-horizon problem), an optimal stopping time is based on a curve that is killed at the moment the mean of the exponential time is reached. That is, an optimal stopping time is the first time the process crosses above a non-negative, continuous, and non-increasing curve depending on time. This curve and the value function are characterised as a solution of a system of nonlinear integral equations which can be understood as a generalisation of the free boundary equations (see e.g. [21, Chapter IV.14.1]) in the presence of jumps. As an example, we numerically calculate this curve in the Brownian motion case and for a compound Poisson process with exponential-sized jumps perturbed by a Brownian motion.

The COVID-19 pandemic remains a public health problem threatening national and global health security. The socio-economic impact of COVID-19 was more severe on developing countries including Lebanon, especially due to the fragile healthcare system, weak surveillance infrastructure and lack of comprehensive emergency preparedness and response plans. Lebanon has been struggling with plethora of challenges at the social, economic, financial, political and healthcare levels prior to the COVID-19 pandemic. The COVID-19 pandemic in Lebanon revealed gaps and challenges across the spectrum of preparedness and response to emergencies. Despite these challenges, the Lebanese response was successful in delaying the steep surge of COVID-19 cases and hospitalisations through imposing strict public health and social measures. The deployment of the national vaccination plan in Lebanon in February 2021 coincided with the reduction in the number of cases and hospitalisation rates. The aim of this manuscript is to advance the epidemiologic evolution of COVID-19 in Lebanon pre- and post-vaccination, the challenges affecting the response and recovery, and the lessons learned.

From the Poisson–Dirichlet diffusions to the Z-measure diffusions, they all have explicit transition densities. We show that the transition densities of the Z-measure diffusions can also be expressed as a mixture of a sequence of probability measures on the Thoma simplex. The coefficients are the same as the coefficients in the Poisson–Dirichlet diffusions. This fact will be uncovered by a dual process method in a special case where the Z-measure diffusions are established through an up–down chain in the Young graph.

We consider solutions of Lévy-driven stochastic differential equations of the form $\textrm{d} X_t=\sigma(X_{t-})\textrm{d} L_t$, $X_0=x$, where the function $\sigma$ is twice continuously differentiable and the driving Lévy process $L=(L_t)_{t\geq0}$ is either vector or matrix valued. While the almost sure short-time behavior of Lévy processes is well known and can be characterized in terms of the characteristic triplet, there is no complete characterization of the behavior of the solution X. Using methods from stochastic calculus, we derive limiting results for stochastic integrals of the form $t^{-p}\int_{0+}^t\sigma(X_{t-})\,\textrm{d} L_t$ to show that the behavior of the quantity $t^{-p}(X_t-X_0)$ for $t\downarrow0$ almost surely reflects the behavior of $t^{-p}L_t$. Generalizing $t^{{\kern1pt}p}$ to a suitable function $f\colon[0,\infty)\rightarrow\mathbb{R}$ then yields a tool to derive explicit law of the iterated logarithm type results for the solution from the behavior of the driving Lévy process.

The invariant Galton–Watson (IGW) tree measures are a one-parameter family of critical Galton–Watson measures invariant with respect to a large class of tree reduction operations. Such operations include the generalized dynamical pruning (also known as hereditary reduction in a real tree setting) that eliminates descendant subtrees according to the value of an arbitrary subtree function that is monotone nondecreasing with respect to an isometry-induced partial tree order. We show that, under a mild regularity condition, the IGW measures are attractors of arbitrary critical Galton–Watson measures with respect to the generalized dynamical pruning. We also derive the distributions of height, length, and size of the IGW trees.

Deferred prosecution agreements (DPAs) are a legal tool for the nontrial resolution of cases of corruption. Each DPA is accompanied by a Statement of Facts that provides detailed and publicly available textual records of the given cases, including summarized evidence of who was involved, what they committed, and with whom. These statements can be translated into networks amenable to social network analysis allowing an analysis of the structure and dynamics of each case. In this study, we show how to extract information about which actors were involved in a given case, the relations and interactions among these actors (e.g., communication or payments), and their relevant individual attributes (gender, affiliation, and sector) from five Statements of Fact. We code the extracted information manually with two independent coders and subsequently, we assess the inter-coder reliability. For assessing the coding reliability of nodes and attributes, we use a matching coefficient, whereas for assessing the coding reliability of ties, we construct a network from the coding of each coder and subsequently calculate the graph correlations of the two resulting networks. The coding of nodes and ties in the five extracted networks turns out to be highly reliable with only slightly lower coding reliability in the case of the largest network. The coding of attributes is highly reliable as well, although it is prone to missing data on actors’ gender. We conclude by discussing the flexibility of our data collection framework and its extension by including network dynamics and nonhuman actors (such as companies) in the network representation.