Optimal embeddings by unbiased shifts of Brownian motion

An unbiased shift of the two-sided Brownian motion $(B_t \colon t\in{\mathbb R})$ is a random time $T$ such that $(B_{T+t} \colon t\in{\mathbb R})$ is still a two-sided Brownian motion. Given a pair $\mu, \nu$ of orthogonal probability measures, an unbiased shift $T$ solves the embedding problem, if $B_0\sim\mu$ implies $B_{T}\sim\nu$. A solution to this problem was given by Last et al. (2014), based on earlier work of Bertoin and Le Jan (1992), and Holroyd and Liggett (2001). In this note we show that this solution minimises ${\mathbb E} \psi(T)$ over all nonnegative unbiased solutions $T$, simultaneously for all nonnegative, concave functions $\psi$. Our proof is based on a discrete concavity inequality that may be of independent interest.