  We study the hedging and valuation of European and American claims on a
non-traded asset $Y$, when a traded stock $S$ is available for hedging, with
$S$ and $Y$ following correlated geometric Brownian motions. This is an
incomplete market, often called a basis risk model. The market agent's risk
preferences are modelled using a so-called forward performance process (forward
utility), which is a time-decreasing utility of exponential type. Moreover, the
market agent (investor) does not know with certainty the values of the asset
price drifts. This market setting with drift parameter uncertainty is the
partial information scenario. We discuss the stochastic control problem
obtained by setting up the hedging portfolio and derive the optimal hedging
strategy. Furthermore, a (dual) forward indifference price representation of
the claim and its PDE are obtained. With these results, the residual risk
process representing the basis risk (hedging error), pay-off decompositions and
asymptotic expansions of the indifference price in the European case are
derived. We develop the analogous stochastic control and stopping problem with
an American claim and obtain the corresponding forward indifference price
valuation formula.
