  We analyze an optimal stopping problem with random maturity under a nonlinear
expectation with respect to a weakly compact set of mutually singular
probabilities $\mathcal{P}$. The maturity is specified as the hitting time to
level $0$ of some continuous index process at which the payoff process is even
allowed to have a positive jump. When $\mathcal{P}$ is a collection of
semimartingale measures, the optimal stopping problem can be viewed as a {\it
discretionary} stopping problem for a player who can influence both drift and
volatility of the dynamic of underlying stochastic flow.
