We study an optimal execution problem in a market model which considers market impact. First we study a discrete-time model and describe a value function. Then, by shortening the intervals of the execution times, we derive the value function of a continuous-time model and study some of its properties (continuity, semi-group property and viscosity property). We show that these vary with the strength of the market impact. We introduce some examples which show that the forms of the optimal strategies change completely, depending on the amount of the trader's security holdings.
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