The original transport problem is to optimally move a pile of soil to an excava-
tion. Mathematically, given two measures of equal mass, we look for an optimal
map that takes one measure to the other one and also minimizes a given cost func-
tional. 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 nal value. Hence, one needs to consider
processes instead of simply the transport maps. The probability distribution of this process
has prescribed marginals at nal 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 joint work with Yan Dolinsky from Hebrew University.