  In this paper we consider an infinite time horizon risk-sensitive optimal
stopping problem for a Feller--Markov process with an unbounded terminal cost
function. We show that in the unbounded case an associated Bellman equation may
have multiple solutions and we give a probabilistic interpretation for the
minimal and the maximal one. Also, we show how to approximate them using finite
time horizon problems. The analysis, covering both discrete and continuous time
case, is supported with illustrative examples.
