Dynamic Programming and Optimal Lookahead Strategies in High Frequency Trading with Transaction Costs

Abstract

An optimal stochastic discrete time control problem with non smooth penalty function is considered. This problem naturally arises in high-frequency trading on financial markets. The principle of dynamic programming is formulated for this problem. Existence and uniqueness of the optimal strategy is proved. An algorithm for building a suboptimal strategy is presented and approximating properties of this strategy are studied. In addition, a simplified strategy is described which is a solution of an isotonic regression problem. A class of problems is defined on which this strategy can replace the basic suboptimal strategy. The results are illustrated by examples.

JEL Classification: C2, C53, C61, F17, F47