  We study an infinite horizon optimal stopping problem which arises naturally
in the optimal timing of a firm/project sale or in the valuation of natural
resources: the functional to be maximised is a sum of a discounted running
reward and a discounted final reward. The running and final rewards as well as
the instantaneous interest rate (used for calculating discount factors) depend
on a Feller-Markov process modelling the underlying randomness in the world,
such as prices of stocks, prices of natural resources (gas, oil, etc.), or
factors influencing the interest rate. For years, it had seemed sensible to
assume that interest rates were uniformly separated from 0, which is needed for
the existing theory to work. However, recent developments in Japan and in
Europe showed that interest rates can get arbitrarily close to 0. In this paper
we establish the feasibility of the stopping problem, prove the existence of
optimal stopping times and a variational characterisation (in the viscosity
sense) of the value function when interest rates are NOT uniformly separated
from 0. Our results rely on certain ergodic properties of the underlying
(non-uniformly) ergodic Markov process. We provide several criteria for
diffusions and jump-diffusions.
