Given an initial (resp., terminal) probability measure μ (resp., ν)
on Rd, we characterize those optimal stopping times Ï„ that
maximize or minimize the functional E∣B0​−Bτ​∣α,
α>0, where (Bt​)t​ is Brownian motion with initial law B0​∼μ
and with final distribution --once stopped at τ-- equal to Bτ​∼ν.
The existence of such stopping times is guaranteed by Skorohod-type
embeddings of probability measures in "subharmoic order" into Brownian motion.
This problem is equivalent to an optimal mass transport problem with certain
constraints, namely the optimal subharmonic martingale transport. Under the
assumption of radial symmetry on μ and ν, we show that the optimal
stopping time is a hitting time of a suitable barrier, hence is non-randomized
and is unique