Probability Commons^™

Population Genetics: Estimation Of Distributions Through Systems Of Non-Linear Differential Equations, Nacer E. Abrouk, Robert J. Lopez

Mathematical Sciences Technical Reports (MSTR)

In stochastic population genetics, the fundamental quantity used for describing the genetic composition of a Mendelian population is the gene frequency. The process of change in the gene frequency is generally modeled as a stochastic process satisfying a stochastic differential equation. The drift and diffusion coefficients in this equation reflect such mechanisms as mutation, selection, and migration that affect the population. Except in very simple cases, it is difficult to determine the probability law of the stochastic process of change in gene frequency. We present a method for obtaining approximations of this process, enabling us to study models more realistic …

Smooth Densities For Degenerate Stochastic Delay Equations With Hereditary Drift, Denis R. Bell, Salah-Eldin A. Mohammed

Articles and Preprints

We establish the existence of smooth densities for solutions of R^d-valued stochastic hereditary differential systems of the form

dx(t) = H(t,x)dt + g(t, x(t - r))dW(t).

In the above equation, W is an n-dimensional Wiener process, r is a positive time delay, H is a nonanticipating functional defined on the space of paths in R^d and g is an n x d matrix-valued function defined on [0, ∞) x R^d, such that gg^* has …