We show that, in a weak sense, as the cost of observation tends to zero, the shape of the continuation region of the Bayes solution for the exponential family problem given above is approximated by that of a corresponding problem for the Wiener process with drift. The approach is an extension of that used in Bickel and Yahav, Proc. Sixth Berkeley Symp. Math. Statist. Prob. (1971).
"On the Asymptotic Shape of Bayesian Sequential Tests of $\theta \leqq 0$ Versus $\theta > 0$ for Exponential Families." Ann. Statist. 1 (2) 231 - 240, March, 1973. https://doi.org/10.1214/aos/1176342361