  We consider the optimal stopping time problem under model uncertainty $R(v)=
{\text{ess}\sup\limits}_{ \mathbb{P} \in \mathcal{P}}
{\text{ess}\sup\limits}_{\tau \in \mathcal{S}_v} E^\mathbb{P}[Y(\tau) \vert
\mathcal{F}_v]$, for every stopping time $v$, set in the framework of families
of random variables indexed by stopping times. This setting is more general
than the classical setup of stochastic processes, and particularly allows for
general payoff processes that are not necessarily right-continuous. Under
weaker integrability, and regularity assumptions on the reward family $Y=(Y(v),
v\in \mathcal{S})$, we show the existence of an optimal stopping time. We then
proceed to find sufficient conditions for the existence of an optimal model.
For this purpose, we present a universal Doob-Meyer-Mertens's decomposition for
the Snell envelope family associated with $Y$ in the sense that it holds
simultaneously for all $\mathbb{P} \in \mathcal{P}$. This decomposition is then
employed to prove the existence of an optimal probability model and study its
properties.
