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In the present paper, we study the optimal execution problem under stochasticprice recovery based on limit order book dynamics. We model price recoveryafter execution of a large order by accelerating the arrival of the refillingorder, which is defined as a Cox process whose intensity increases by thedegree of the market impact. We include not only the market order but also thelimit order in our strategy in a restricted fashion. We formulate the problemas a combined stochastic control problem over a finite time horizon. Thecorresponding Hamilton-Jacobi-Bellman quasi-variational inequality is solvednumerically. 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 bythe tolerance for market impact depending on the state at each time step, andhence the strategy behaves dynamically. We also provide competitive results dueto inclusion of the limit order, even though a limit order is allowed underconservative evaluation of the execution price.

A dynamic optimal execution strategy under stochastic price recovery