Summary: Suppose that \(X=(X_ t,F_ t, t\in R_ +)\) is an optional reward process with \((F_ t)\) satisfying usual conditions. We correct the proof of existence about Snell envelope in [J.-F. Mertens, Z. Wahrscheinlichkeitstheorie Verw. Geb. 22, 45-68 (1972; Zbl 0236.60033)] and the proof of an important lemma (Lemma 4.6) in [M. E. Thompson, ibid. 19, 302-318 (1971; Zbl 0208.439)] and give a proof of the existence about Snell envelope under certain conditions, i.e. \(EZ^ -_ \infty<\infty\) and \(Z\) is upper-semi-continuous on the right or there is a stopping rule \(\tau\geq\sigma\) such that \(EZ^ -_ \tau<\infty\) for any stopping rule \(\sigma\). At the same time, we prove a four-repeated limit theorem when \(Z\) is continuous on the right. The character and the uniqueness of the optimal stopping time or optimal stopping rule are discussed.