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.