Weak Approximations of the Wright–Fisher Process

In this paper, we construct first- and second-order weak split-step approximations for the solutions of the Wright–Fisher equation. The discretization schemes use the generation of, respectively, two- and three-valued random variables at each discretization step. The accuracy of constructed approximations is illustrated by several simulation examples.

The Rescaled Pólya Urn and the Wright–Fisher Process with Mutation

In recent papers the authors introduce, study and apply a variant of the Eggenberger–Pólya urn, called the “rescaled” Pólya urn, which, for a suitable choice of the model parameters, exhibits a reinforcement mechanism mainly based on the last observations, a random persistent fluctuation of the predictive mean and the almost sure convergence of the empirical mean to a deterministic limit. In this work, motivated by some empirical evidence, we show that the multidimensional Wright–Fisher diffusion with mutation can be obtained as a suitable limit of the predictive means associated to a family of rescaled Pólya urns.

On the Simpson index for the Wright–Fisher process with random selection and immigration

Moran or Wright–Fisher processes are probably the most well known models to study the evolution of a population under environmental various effects. Our object of study will be the Simpson index which measures the level of diversity of the population, one of the key parameters for ecologists who study for example, forest dynamics. Following ecological motivations, we will consider, here, the case, where there are various species with fitness and immigration parameters being random processes (and thus time evolving). The Simpson index is difficult to evaluate when the population is large, except in the neutral (no selection) case, because it has no closed formula. Our approach relies on the large population limit in the “weak” selection case, and thus to give a procedure which enables us to approximate, with controlled rate, the expectation of the Simpson index at fixed time. We will also study the long time behavior (invariant measure and convergence speed towards equilibrium) of the Wright–Fisher process in a simplified setting, allowing us to get a full picture for the approximation of the expectation of the Simpson index.