  We study a multiplicative transient price impact model for an illiquid
financial market, where trading causes price impact which is multiplicative in
relation to the current price, transient over time with finite rate of
resilience, and non-linear in the order size. We construct explicit solutions
for the optimal control and the value function of singular optimal control
problems to maximize expected discounted proceeds from liquidating a given
asset position. A free boundary problem, describing the optimal control, is
solved for two variants of the problem where admissible controls are monotone
or of bounded variation.
