Optimal stopping under <i>G</i>-expectation

In this study, we develop a theory of optimal stopping problems within the G-expectation framework. To address this problem, we first introduce a type of random times, called G-stopping times, which are specifically suited for this setting. In the discrete-time case with a finite horizon, we define the value function backward and show that it is the smallest G-supermartingale that dominates the payoff process, ensuring the existence of an optimal stopping time. We then extend these results to both the infinite-horizon case and the continuous-time setting. Moreover, we establish the relationship between the value function and the solution of the reflected backward stochastic differential equation driven by G-Brownian motion.