In this paper we consider a new kind of optimal stopping problems. Let {x_n,y_n,ζ_n}^∞_n=1 be an integrable and adapted stochastic process. We will find an optimal stopping rule for {x_n, ζ_n}^∞_n=1 in the class of stopping rule D such that for any t∈D，Ey_t≥V_y-a， where α is a constant and V_y is the value of {y_n, ζ_n}^∞_n=1 such that V_y＜∞.Adopting the Lagrange's method and the generalized Snell's method we obtain some existence theorems respectively，Finally，we discuss the two methods and apply them to solve the optimal stopping problem of the stochastic sequence of randon vectors.