Sequential Variational Testing Hypotheses on the Wiener Process Under Delayed Observations

This paper proposes a sequential procedure for testing the null hypothesis absence of a drift in the Wiener process against the hypothesis that the drift is present. The decision is taken on the basis of current and randomly delayed observations. For any given error probabilities, the optimal decision rule minimizes a certain convex combination of the mean durations corresponding to these hypotheses. By a special geometric result, a convex characterization of the risk is obtained, and the initial problem is reduced to the optimal stopping problem.