Summary: A general problem of testing two simple hypotheses about the distribution of a discrete-time stochastic process is considered. The main goal is to minimize an average sample number over all sequential tests whose error probabilities do not exceed some prescribed levels. As a criterion of minimization, the average sample number under a third hypothesis is used [modified Kiefer-Weiss problem; {\it J. Kiefer} and {\it L. Weiss}, Ann. Math. Stat. 28, 57--74 (1957;

Zbl 0079.35406)]. For a class of sequential testing problems, the structure of optimal sequential tests is characterized. An application to the Kiefer-Weiss problem for discrete-time stochastic processes is proposed. As another application, the structure of Bayes sequential tests for two composite hypotheses, with a fixed cost per observation, is given. The results are also applied for finding optimal sequential tests for discrete-time Markov processes. In a particular case of testing two simple hypotheses about a location parameter of an autoregressive process of order 1, it is shown that the sequential probability ratio test has the {\it A. Wald} and {\it J. Wolfowitz} optimality property [Ann. Math. Stat. 19, 326--339 (1948;

Zbl 0032.17302)].