  In the frictionless discrete time financial market of Bouchard et al.(2015)
we consider a trader who, due to regulatory requirements or internal risk
management reasons, is required to hedge a claim $\xi$ in a risk-conservative
way relative to a family of probability measures $\mathcal{P}$. We first
describe the evolution of $\pi_t(\xi)$ - the superhedging price at time $t$ of
the liability $\xi$ at maturity $T$ - via a dynamic programming principle and
show that $\pi_t(\xi)$ can be seen as a concave envelope of $\pi_{t+1}(\xi)$
evaluated at today's prices. Then we consider an optimal investment problem for
a trader who is rolling over her robust superhedge and phrase this as a robust
maximisation problem, where the expected utility of inter-temporal consumption
is optimised subject to a robust superhedging constraint. This utility
maximisation is carrried out under a new family of measures $\mathcal{P}^u$,
which no longer have to capture regulatory or institutional risk views but
rather represent trader's subjective views on market dynamics. Under suitable
assumptions on the trader's utility functions, we show that optimal investment
and consumption strategies exist and further specify when, and in what sense,
these may be unique.
