Sharp approximations of ruin probabilities in the discrete time models

According to Solvency II directive, each insurance company could determine solvency capital requirements using its own, tailor made, internal model. This highlights the urgency of having fast numerical tools based on practically-oriented mathematical models. From the Solvency II perspective discrete time framework seems to be the most relevant one. In this paper, we propose a number of fast and accurate approximations of ruin probabilities involving some integral operator and examine them along strictly theoretical as well as numerical lines. For a few claim distributions the approximations are shown to be exact. In general, we prove that they converge with an exponential rate to the exact ruin probabilities without any restrictive assumptions on the claim distribution. A fast algorithm to approximate ruin probabilities by a numerical fixed point of the involved integral operator is given. As an application, ruin probabilities for, e.g. normally and Weibull – distributed claims are computed. Comparisons with discrete time counterparts of some continuous time approximation methods are also carried out. Numerical studies show that our approximations are precise both for small and large values of the initial surplus u. In contrast, the empirical De Vylder-type ones strongly depend on the claim distributions and are less precise for small and medium values of u.     

The growth of patenting in New Zealand, 1860–99

Patent applications by male New Zealand inventors sharply increased in the early 1880s after initial official fees were reduced, and the requirement to advertise applications in newspapers abolished. Increasingly, however, applications lapsed, while applications by unskilled workers remained low. Non-fee costs were crucially important, with the 1870 reduction in fees failing to increase patenting, as hoped, because the doubling of mandatory advertising costs negated the fees reduction. Patenting by overseas inventors was less affected by fees, and steadily grew. Only one application was by an indigenous Māori person, while even in 1899 women made just 2.5% of applications.     

A pulse check on recent developments in time series econometrics

This article motivates and summarizes the contributions of the special issues on Recent Developments in Time Series Econometrics.