  We consider a stochastic logistic growth model involving both birth and death
rates in the drift and diffusion coefficients for which extinction eventually
occurs almost surely. The associated complete Fokker-Planck equation describing
the law of the process is established and studied. We then use its solution to
build a likelihood function for the unknown model parameters, when discretely
sampled data is available. The existing estimation methods need adaptation in
order to deal with the extinction problem. We propose such adaptations, based
on the particular form of the Fokker-Planck equation, and we evaluate their
performances with numerical simulations. In the same time, we explore the
identifiability of the parameters which is a crucial problem for the
corresponding deterministic (noise free) model.
