  In practice, one must recognize the inevitable incompleteness of information
while making decisions. In this paper, we consider the optimal redeeming
problem of stock loans under a state of incomplete information presented by the
uncertainty in the (bull or bear) trends of the underlying stock. This is
called drift uncertainty. Due to the unavoidable need for the estimation of
trends while making decisions, the related Hamilton-Jacobi-Bellman (HJB)
equation is of a degenerate parabolic type. Hence, it is very hard to obtain
its regularity using the standard approach, making the problem different from
the existing optimal redeeming problems without drift uncertainty. We present a
thorough and delicate probabilistic and functional analysis to obtain the
regularity of the value function and the optimal redeeming strategies. The
optimal redeeming strategies of stock loans appear significantly different in
the bull and bear trends.
