Stack inspection is a key technology for runtime access control of programs in a network environment. In this paper, a verification problem to decide whether a given program with stack inspection satisfies a given security property is discussed. First, the computational complexity of the problem is investigated. Since the result implies the problem is computationally intractable in general, we introduce a practically important subclass of programs which exactly model programs containing check Permission of Java development kit 1.2. We show that the problem for this subclass is solvable in linear time in the size of a program.