Abstract: In this paper, it is researched that the structure of discrete two indexes stochastic processes X=（X_z,F_z,z∈N²） and the asymptotic arithmetic of Shell’s envelope of X. At first, under the condition （A⁺）, it is proved that Γ=（γ_z,F_z,z∈N²） is minimal F-regul alsupermatingale above X, wherer_n=\underset\sigma \in \Sigma, \sigma>\mathbbF\operatornameess \sup E\left(X_\sigma \mid \mathscrF_n\right). Then optimal tactics is structured by Snell’s envelope of X and optimal principle. Therfore, the structure of optimal stopping for payoff processes X=（X_z,F_z,z∈N²） are obtained. At last, three limit theorem for Snell’s envelope of X is proved.