Continuous state branching processes in random environment: The Brownian case

We consider continuous state branching processes that are perturbed by a Brownian motion. These processes are constructed as the unique strong solution of a stochastic differential equation. The long-term extinction and explosion behaviours are studied. In the stable case, the extinction and explosion probabilities are given explicitly. We find three regimes for the asymptotic behaviour of the explosion probability and, as in the case of branching processes in random environment, we find five regimes for the asymptotic behaviour of the extinction probability. In the supercritical regime, we study the process conditioned on eventual extinction where three regimes for the asymptotic behaviour of the extinction probability appear. Finally, the process conditioned on non-extinction and the process with immigration are given.Comment: New version, Theorem 1 is improved and a new regime appears in the supercritical cas

Database of historic ports and coastal sailing routes in England and Wales

This data paper presents a reconstruction of historical ports and coastal routes in England and Wales during the age of the sailing ship, ending at the beginning of the twentieth century. The dataset was created by an amalgamation of twenty different sources,including geographical data, primary sources and secondary literature. Ports found in historical documents were listed by year of appearance and georeferenced. Ports that appear in multiple sour-ces were listed only once. Coastal routes between ports were drawn based on navigation charts and bathymetry data, distinguishing five categories with different characteristics. Visibility from the coast was deduced from elevation rasters and lighthouse locations. Subsequently both ports and coastal routes were checked using topo-logical rules to ensure the connectivity of the network. The data is provided in shapefile format with all the attributes and can be analysed using Geographical Information Systems (GIS) for different types of geographical and historical studies

Asymptotic behavior of projections of supercritical multi-type continuous state and continuous time branching processes with immigration

Under a fourth order moment condition on the branching and a second order moment condition on the immigration mechanisms, we show that an appropriately scaled projection of a supercritical and irreducible continuous state and continuous time branching process with immigration on certain left non-Perron eigenvectors of the branching mean matrix is asymptotically mixed normal. With an appropriate random scaling, under some conditional probability measure, we prove asymptotic normality as well. In case of a non-trivial process, under a first order moment condition on the immigration mechanism, we also prove the convergence of the relative frequencies of distinct types of individuals on a suitable event; for instance, if the immigration mechanism does not vanish, then this convergence holds almost surely.Comment: 53 page

Asymptotic behaviour of exponential functionals of L\'evy processes with applications to random processes in random environment

Let $\xi=(\xi_t, t\ge 0)$ be a real-valued L\'evy process and define its associated exponential functional as follows $$I_t(\xi):=\int_0^t \exp\{-\xi_s\}{\rm d} s, \qquad t\ge 0.$$ Motivated by important applications to stochastic processes in random environment, we study the asymptotic behaviour of $$\mathbb{E}\Big[F\big(I_t(\xi)\big)\Big] \qquad \textrm{as}\qquad t\to \infty,$$ where $F$ is a non-increasing function with polynomial decay at infinity and under some exponential moment conditions on $\xi$. In particular, we find five different regimes that depend on the shape of the Laplace exponent of $\xi$. Our proof relies on a discretisation of the exponential functional $I_t(\xi)$ and is closely related to the behaviour of functionals of semi-direct products of random variables. We apply our main result to three {questions} associated to stochastic processes in random environment. We first consider the asymptotic behaviour of extinction and explosion for {stable} continuous state branching processes in a L\'evy random environment. Secondly, we {focus on} the asymptotic behaviour of the mean of a population model with competition in a L\'evy random environment and finally, we study the tail behaviour of the maximum of a diffusion process in a L\'evy random environment.Comment: arXiv admin note: text overlap with arXiv:1512.07691, arXiv:math/0511265 by other authors. Results are improve

How to Integrate Disability Benefits into a System with Individual Accounts: The Chilean Model

Chile offers an innovative approach to disability insurance within a multi-pillar social security system. The individual’s retirement savings account is used as part of his disability insurance, but if he becomes disabled the account is topped up enough to finance a defined benefit annuity that pays 70% of the reference wage. This is accomplished primarily through the private insurance market, but with government providing regulations and back-up guarantees. The private pension funds and insurance companies that participate in the assessment process have a pecuniary interest in keeping costs low, unlike most public systems that are run by agencies without a personal incentive to contain costs. The individual accounts, used for old age retirement savings, help to finance the disability and survivors’ insurance as a joint product. This further keeps costs down and makes the system less sensitive to demographic shocks than a public pay-as-you-go system would be. However, pre-funding a defined benefit makes system costs much more sensitive to interest rate shifts. The defined benefit reduces risk to the worker but non-differentiated pricing creates cross-subsidies and, in a competitive market, incentives for creaming. Some of the cost reductions to the private insurance may imply a larger future public obligation, due to the minimum pension guarantee. This study examines the potential successes and pitfalls of this mixed public-private funded system of disability insurance and evaluates whether it provides a useful model for the US and other countries.