We consider the optimal structure of couplings for the Martingale transport problem with initial and terminal laws \mu, \nu on R^d \R . Here optimality means that such solutions will minimize the functional E \E |X-Y|^p . We show how such optimal couplings are described and unique when 0<p %DIFDELCMD < \leq1%%% . We apply the monotonicity principle introduced in 4 , and the assumption of radial symmetry of \mu and \nu is important .
We determine the optimal structure of couplings for the Martingale transport problem between radially symmetric initial and terminal laws \mu, \nu on \R^d and show the uniqueness of optimizer . Here optimality means that such solutions will minimize the functional \E |X-Y|^p where 0<p %DIFDELCMD < \leq1%%% \leq 1 , and the dimension d is arbitrary .
R
consider
determine
R
with
between radially symmetric
D
R^d
A
^d and show the uniqueness of optimizer
D
E
R
. We show how such optimal couplings are described and unique when
where
D
. We apply the monotonicity principle introduced in
R
4
\leq 1
R
assumption of radial symmetry of \mu and \nu is important
dimension d is arbitrary