The martingale optimal transportation problem is motivated by
model-independent bounds for the pricing and hedging exotic options in
financial mathematics.
In the simplest one-period model, the dual formulation of the robust
superhedging cost differs from the standard optimal transport problem by
the presence of a martingale constraint on the set of coupling measures.
The one-dimensional Brenier theorem has a natural extension. However, in
the present martingale version, the optimal coupling measure is
concentrated on a pair of graphs which can be obtained in explicit form.
These explicit extremal probability measures are also characterized as
the unique left and right monotone martingale transference plans, and
induce an optimal solution of the kantorovitch dual, which coincides
with our original robust hedging problem.
By iterating the above construction over n steps, we define a Markov
process whose distribution is optimal for the n-periods martingale
transport problem corresponding to a convenient class of cost functions.
Similarly, the optimal solution of the corresponding robust hedging
problem is deduced in explicit form. Finally, by sending the time step
to zero, this leads to a continuous-time version of the one-dimensional
Brenier theorem in the present martingale context, thus providing a new
remarkable example of Peacock, i.e. Processus Croissant pour l'Ordre
Convexe. Here again, the corresponding robust hedging strategy is
obtained in explicit form.