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LERCHE'S SEQUENTIAL TEST FOR THE DRIFT OF A BROWNIAN MOTION WITH A SMOOTH PRIOR

Lerche (1986) studies a specific sequential problem concerned with testing the sign of the drift (say θ) of a Brownian motion. He shows that a repeated significance test boundary based on frequentist principles is also an optimal Bayesian boundary with 0–1 decision loss and sampling costs cθ2 per unit time, where c > 0. In this paper, we re-investigate Lerche's problem for a more general smooth prior. Using the approach developed by Simons et al. (1989), we derive an expansion of the optimal stopping boundary by solving a free boundary problem for the heat equation. The first term of the solution of the free boundary problem is exactly the same as for Lerche's optimal stopping boundary, while the first term of the corresponding Bayes risk is slightly different from Lerche's Bayes risk. An alternative approach to approximating the boundary can be obtained by using a tangent approximation for two-sided Brownian exit densities. Doing so yields results that are close to the results obtained by solving the free problem.