On the risk of extinction for a population subject to a random Markov evolution with jumps

We consider a one-dimensional population whose evolution is described by a jump-diffusion equation and we study the effects of changing the coefficients of the equation on the extinction time, that is the instant at which the population becomes arbitrarily small. It is shown that, under the same diffusion coefficient, if one reduces the drift and the size of jumps, the speed of extinction increases; moreover, the probability of reaching a higher population state than the present one before reaching a lower population size decreases. If the diffusion coefficient is state-independent, the speed of extinction increases with it. Furthermore, if no jumps are allowed (i.e. for a simple-diffusion equation), then under certain conditions on the coefficients of the equation both large and small values of the diffusion coefficient result in a higher extinction risk.