Extended Conditional G-Expectations and Related Stopping Times

In this paper we extend the definition of time conditional G-expectations $\mathbb{\hat{E}}_{t}[\cdot]$ to a larger domain on which the dynamical consistency still holds. In fact we can consistently define, by taking the limit, the time conditional expectations for each random variable $X$ which is the downward limit (resp. upward limit) of a monotone sequence $\{X_{i}\}$ in $L_{G}^{1}(\Omega)$. To accomplish this procedure, some careful analysis is needed. Moreover, we give a suitable definition of stopping times and obtain the optional stopping theorem. We also provide some basic and interesting properties for the extended conditional G-expectations.