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An unbiased shift of the two‐sided Brownian motion    (   B  t   :  t  ∈  R  )    is a random time   T   such that    (   B   T  +  t    :  t  ∈  R  )    is still a two‐sided Brownian motion. Given a pair    μ  ,  ν    of orthogonal probability measures, an unbiased shift   T   solves the embedding problem, ifB  0   ∼  μ    impliesB  T   ∼  ν   . 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    E  ψ  (  T  )    over all non‐negative unbiased solutions   T  , simultaneously for all non‐negative, concave functions   ψ  . Our proof is based on a discrete concavity inequality that may be of independent interest.

Optimal embeddings by unbiased shifts of Brownian motion