Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5130048 | Stochastic Processes and their Applications | 2017 | 44 Pages |
We analyze an optimal stopping problem supγâTE¯0[Yγâ§Ï0] with random maturity Ï0 under a nonlinear expectation E¯0[â ]:=supPâPEP[â ], where P is a weakly compact set of mutually singular probabilities. The maturity Ï0 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 (Pâ,γâ) for sup(P,γ)âPÃTEP[Yγâ§Ï0], in which γâ is the first time Y meets the limit Z of its approximating E¯âSnell 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 Ï0 by an increasing sequence of Lipschitz continuous stopping times and approximate Y by a sequence of uniformly continuous processes.