Before revision:  We consider a trader who wishes to liquidate a large amount of currency, where for whom prices are optimal at the barrier. The central bank, who wishes to keep the currency exchange rate above this barrier, therefore needs to buy its own currency. The permanent price impact, which is created by the transactions of both sides, turns the optimal trading problems of the trader and the central bank into coupled singular control problems, where the common singularity arises from a local time along a random curve. We first solve the central bank's control problem by means of the Skorokhod map and then derive the trader's optimal strategy by solving a sequence of approximated control problems, thus establishing a Stackelberg equilibrium in our model .
After revision:  We consider a trader who wishes to maximize the expected local time through generating price impact. The local time can be regarded as a proxy for the inventory of a central bank whose aim is to maintain a target zone .