  In the present paper, we study the optimal execution problem under stochastic
price recovery based on limit order book dynamics. We model price recovery
after execution of a large order by accelerating the arrival of the refilling
order, which is defined as a Cox process whose intensity increases by the
degree of the market impact. We include not only the market order but also the
limit order in our strategy in a restricted fashion. We formulate the problem
as a combined stochastic control problem over a finite time horizon. The
corresponding Hamilton-Jacobi-Bellman quasi-variational inequality is solved
numerically. The optimal strategy obtained consists of three components: (i)
the initial large trade; (ii) the unscheduled small trades during the period;
(iii) the terminal large trade. The size and timing of the trade is governed by
the tolerance for market impact depending on the state at each time step, and
hence the strategy behaves dynamically. We also provide competitive results due
to inclusion of the limit order, even though a limit order is allowed under
conservative evaluation of the execution price.
