Extended conditional G-expectations and related stopping times

In this paper, we extend the definition of conditional \begin{document}$ G{\text{-}}{\rm{expectation}} $\end{document} to a larger space on which the dynamical consistency still holds. We can consistently define, by taking the limit, the conditional \begin{document}$ G{\text{-}}{\rm{expectation}} $\end{document} for each random variable \begin{document}$ X $\end{document} , which is the downward limit (respectively, upward limit) of a monotone sequence \begin{document}$ \{X_{i}\} $\end{document} in \begin{document}$ L_{G}^{1}(\Omega) $\end{document} . To accomplish this procedure, some careful analysis is needed. Moreover, we present a suitable definition of stopping times and obtain the optional stopping theorem. We also provide some basic and interesting properties for the extended conditional \begin{document}$ G{\text{-}}{\rm{expectation}} $\end{document} .