A martingale approach for detecting the drift of a Wiener process (Q1593617)

The sequential Bayes-test problem for unknown drift \(\theta \in R\) of the Wiener process \((W_t)_{t\geq 0}\) is considered. \textit{H. R. Lerche} [Ann. Stat. 14, 1030-1048 (1986; Zbl 0604.62078)] showed that in the case of a normal prior an \(o(c)\)-optimal sequential probability test (SPRT) \(T_{b(c)}\) could be constructed, when according to the cost constant \(c\) the level \(b(c)\) is appropriately chosen. In this paper a new martingale approach is introduced. The representation of the density process \((f(W_t,t))_{t\geq 0}\) as exponential martingale together with some facts on one-sided SPRTs provide the key formula \[ \int \theta ^2E_{\theta}T_bF(d\theta)=2\log b + \overline {E}\int _0^{T_{b}} v(W_s,s) ds, \] \(v\) denoting the variance of the posterior distribution, which allows an asymptotic expansion for \(b\) tending to infinity. Relations to the optimal Bayes risk are given, which show the \(o(c)\)-optimality for suitable nonnormal priors.