Establish convexity of the relevant payoff functions in the Brownian-drift stopping problem

Establish convexity of the relevant payoff functions needed to recover the time-monotonicity result of Ekström and Wang for the Brownian motion with unknown drift model.

Background

For a Brownian motion with unknown drift, the paper identifies the posterior statistic Pi_t^T as a martingale diffusion whose diffusion coefficient is the posterior covariance Cov(T(Theta),Theta\mid\mathcal F_t^Y). The paper proves that this coefficient decreases over time along posterior level curves and establishes convexity preservation for posterior transition operators under its general assumptions.

The authors explicitly note that these results do not fully recover the earlier time-monotonicity theorem of Ekström and Wang because the convexity of the payoff functions relevant to that stopping problem remains unestablished. Resolving this issue would complete the connection between the diffusion-coefficient monotonicity and the desired monotonicity of the value function and stopping boundaries.