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.

翻译：Gapeev-Shiryaev猜想（源自Gapeev与Shiryaev（2011）及Gapeev与Shiryaev（2013）的研究）可概括表述为：信噪比的单调性蕴涵最优停时边界的单调性。该猜想最初在两类扩散过程问题中提出：（i）序贯检验问题（需判断两个漂移项中哪一个被间接观测）；（ii）最快检测问题（需检测初始漂移何时变为新漂移）。本文分别在序贯检验框架（基于底层随机微分方程的Lipschitz/Hölder系数）与最快检测框架（基于底层随机微分方程的解析系数）下给出Gapeev-Shiryaev猜想的证明。序贯检验框架的证明方法依赖于随机时间变换与轨道比较论证，而最快检测框架中这两种论证均失效，需代之以满足Hörmander条件的次椭圆方程的随机最大值原理（该原理本身具有独立研究价值）。Gapeev-Shiryaev猜想的验证确立了如下事实：具有单调信噪比的序贯检验与最快检测问题可通过已知方法求解。