  The second fundamental theorem of asset pricing characterizes completeness of
a financial market by uniqueness of prices of financial claims. The associated
super- and sub-hedging dualities give upper and lower bounds of the
no-arbitrage interval. In this article we provide conditional versions of these
results in discrete time.
  The main tool we use are consistency properties of dynamic non-linear
expectations, which we apply to the super- and sub-hedging prices. The obtained
results extend existing results in the literature, where the conditional
setting is in most cases considered only on finite probability spaces. Using
non-linear expectations we also provide a new perspective on the optional
decomposition theorem.
