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Stochastic differential equations are often simulated with the Monte Carlo Euler method. Convergence of this method is well understood in the case of globally Lipschitz continuous coefficients of the stochastic differential equation. The important case of superlinearly growing coefficients, however, remained an open question for a long time now. The main… (More)

The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation with globally Lipschitz coefficients and even with coefficients which grow at most linearly. For super-linearly growing coefficients convergence in the strong and numerically weak sense remained an open question. In this article we prove for many… (More)

A continuous mass population model with local competition is constructed where every emigrant colonizes an unpopulated island. The population founded by an emigrant is modeled as excursion from zero of an one-dimensional diffusion. With this excursion measure, we construct a process which we call Virgin Island Model. A necessary and sufficient condition for… (More)

For a class of processes modeling the evolution of a spatially structured population with migration and a logistic local regulation of the reproduction dynamics we show convergence towards an upper invariant measure from a suitable class of initial distributions. It follows from recent work of A. Etheridge that this upper invariant measure is non-trivial… (More)

Species introductions to new habitats can cause a decline in the population size of competing native species and consequently also in their genetic diversity. We are interested in why these adverse effects are weak in some cases whereas in others the native species declines to the point of extinction. While the introduction rate and the growth rate of the… (More)

J o u r n a l o f P r o b a b i l i t y Electron. Abstract We consider systems of interacting diffusions with local population regulation representing populations on countably many islands. Our main result shows that the total mass process of such a system is bounded above by the total mass process of a tree of excursions with appropriate drift and… (More)

- MARTIN HUTZENTHALER, JESSE EARL TAYLOR, HUTZENTHALERAND J. E. TAYLOR
- 2010

We describe the processes obtained by time reversal of a class of stationary jump diffusion processes that model the dynamics of genetic variation in populations subject to repeated bottlenecks. Assuming that only one lineage survives each bottleneck, the forward process is a diffusion on [0, 1] that jumps to the boundary before diffusing back into the… (More)

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