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

Optimal redeeming strategy of stock loans under drift uncertainty