The original transport problem is to optimally move a pile of soil to an excavation.
Mathematically, given two measures of equal mass, we look for an optimal bijection that takes
one measure to the other one and also minimizes a given cost functional. Kantorovich relaxed
this problem by considering a measure whose marginals agree with given two measures instead of
a bijection. This generalization linearizes the problem. Hence, allows for an easy existence
result and enables one to identify its convex dual.
In robust hedging problems, we are also given two measures. Namely, the initial and the final
distributions of a stock process. We then construct an optimal connection. In general, however,
the cost functional depends on the whole path of this connection and not simply on the final value.
Hence, one needs to consider processes instead of simply the maps S. The probability distribution
of this process has prescribed marginals at final and initial times. Thus, it is in direct analogy
with the Kantorovich measure. But, financial considerations restrict the process to be a martingale
Interestingly, the dual also has a financial interpretation as a robust hedging (super-replication)
problem.
In this talk, we prove an analogue of Kantorovich duality: the minimal super-replication cost in
the robust setting is given as the supremum of the expectations of the contingent claim over all
martingale measures with a given marginal at the maturity.
This is joint work with Yan Dolinsky of Hebrew University.