Optimal stopping with random maturity under nonlinear expectations

We analyze an optimal stopping problem supγ∈TE¯0[Yγ∧τ0] with random maturity τ0 under a nonlinear expectation E¯0[⋅]:=supP∈PEP[⋅], where P is a weakly compact set of mutually singular probabilities. The maturity τ0 is specified as the hitting time to level 0 of some continuous index process X at which the payoff process Y is even allowed to have a positive jump. When P collects a variety of semimartingale measures, the optimal stopping problem can be viewed as a discretionary stopping problem for a player who can influence both drift and volatility of the dynamic of underlying stochastic flow.We utilize a martingale approach to construct an optimal pair (P∗,γ∗) for sup(P,γ)∈P×TEP[Yγ∧τ0], in which γ∗ is the first time Y meets the limit Z of its approximating E¯−Snell envelopes. To overcome the technical subtleties caused by the mutual singularity of probabilities in P and the discontinuity of the payoff process Y, we approximate τ0 by an increasing sequence of Lipschitz continuous stopping times and approximate Y by a sequence of uniformly continuous processes.