The Moments of Stochastic Integrals and the Distribution of Sojourn Times

摘要

For a single diallelic locus in a finite population with any time-independent selection scheme, using the diffusion approximation, a formula is derived in terms of sojourn times for the moments of the integral of an arbitrary function of gene frequency along sample paths. Irreversible mutation and conditioned and unconditioned processes without mutation are treated. From this expression, the differential equation satisfied by the moments follows directly, and the exact probability distribution of sojourn times is deduced. An independent probabilistic proof of the last result based on the properties of time-homogeneous Markov processes is presented.