Optimal execution problem: a combined stochastic optimal control approach

Abstract

We propose the optimal execution strategy including the limit order. The arrival of the counterpart market order of our limit order is modelled by the Poisson process reduced from the Cox process. We formulate the limit order and market order by the regular stochastic control and impulse control respectively. The performance criterion consists of the terminal wealth and the inventory penalty function which copes with the uncertainty of the limit order. We solve the corresponding Hamilton-Jacobi-Bellman quasi variational inequality numerically using the implicit method. The result of the numerical simulation with the obtained optimal strategy gives a quantitative evaluation of the advantage to include the limit order into the execution strategy.