The Convergence Rate of Stochastic Tracking with Application to Optimal Execution

Abstract

We study the quadratic tracking problem of a general stochastic target process with absolutely continuous controls, with and without terminal constraint. We derive explicit, non-asymptotic upper bounds in terms of a Besov-type modulus of the target. These bounds yield sharp explicit rates that specialize to the square-root order for semimartingale targets. We then apply these results to a generalized Obizhaeva--Wang execution model with random terminal inventory. We first develop a Hilbert-space approach to characterize its optimal strategy, which includes jumps. To avoid such trading spikes, one regularizes the problem by a quadratic trading-rate penalty with coefficient $\varepsilon$. We then show that the regularized optimal execution cost---and therefore the excess price impact cost of the regularized optimal strategy---converges at the sharp rate $O(\sqrt{\varepsilon})$. Since the regularized optimal strategy is not available in closed form, we further construct a nearly optimal strategy which is readily implementable and shares the same approximation rate.