For radially symmetric μ, ν in d≥3 with α≠2, optimal stopping times maximizing/minimizing E[|B0−Bτ|^α] are unique non-randomized hitting times to symmetric barriers.
Hobson, Robust hedging of the lookback option, Finance and Stochastics, 2 (1998) 329–347
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Optimal Brownian stopping when the source and target are radially symmetric distributions
For radially symmetric μ, ν in d≥3 with α≠2, optimal stopping times maximizing/minimizing E[|B0−Bτ|^α] are unique non-randomized hitting times to symmetric barriers.