  The Gapeev-Shiryaev conjecture (originating in Gapeev and Shiryaev (2011) and
Gapeev and Shiryaev (2013)) can be broadly stated as follows: Monotonicity of
the signal-to-noise ratio implies monotonicity of the optimal stopping
boundaries. The conjecture was originally formulated both within (i) sequential
testing problems for diffusion processes (where one needs to decide which of
the two drifts is being indirectly observed) and (ii) quickest detection
problems for diffusion processes (where one needs to detect when the initial
drift changes to a new drift). In this paper we present proofs of the
Gapeev-Shiryaev conjecture both in (i) the sequential testing setting (under
Lipschitz/Holder coefficients of the underlying SDEs) and (ii) the quickest
detection setting (under analytic coefficients of the underlying SDEs). The
method of proof in the sequential testing setting relies upon a stochastic time
change and pathwise comparison arguments. Both arguments break down in the
quickest detection setting and get replaced by arguments arising from a
stochastic maximum principle for hypoelliptic equations (satisfying Hormander's
condition) that is of independent interest. Verification of the Gapeev-Shiryaev
conjecture establishes the fact that sequential testing and quickest detection
problems with monotone signal-to-noise ratios are amenable to known methods of
solution.
