OPTIMAL BROWNIAN STOPPING WHEN THE SOURCE AND TARGET ARE RADIALLY SYMMETRIC DISTRIBUTIONS

Given two probability measures mu, nu on R-d, in subharmonic order, we describe optimal stopping times tau that maximize/minimize the cost functional E vertical bar B-0 - B-tau vertical bar(alpha), alpha > 0, where (B-t)(t) is Brownian motion with initial law mu and with final distribution-once stopped at tau-equal to nu. Under the assumption of radial symmetry on mu and nu, we show that in dimension d >= 3 and alpha not equal 2, there exists a unique optimal solution given by a nonrandomized stopping time characterized as the hitting time to a suitably symmetric barrier. We also relate this problem to the optimal transportation problem for subharmonic martingales and establish a duality result.