  Consider the sample path of a one-dimensional diffusion for which the
diffusion coefficient is given and where the drift may take on one of two
values: $\mu_0$ or $\mu_1$. Suppose that the signal-to-noise ratio (defined as
the difference between the two possible drifts divided by the diffusion
coefficient) is non-constant. Given an initial state for the observed process,
we consider a minimax formulation of the Wiener sequential testing problem for
detecting the correct drift coefficient as soon as possible and with minimal
probabilities of incorrect terminal decisions. We solve the problem in the
Bayesian formulation, under any prior probabilities of the process having drift
$\mu_0$ or $\mu_1$, when the passage of time is penalized linearly. In the case
where the signal-to-noise ratio is assumed constant, we obtain an explicit
formula for the least favorable distribution.
