A Partitioning Algorithm for Markov Decision Processes with Applications to Market Microstructure

We propose a partitioning algorithm to solve a class of linear-quadratic Markov decision processes with inequality constraints and nonconvex stagewise cost; within each region of the partitioned state space, the value function and the optimal policy have analytical quadratic and linear forms, respectively. Compared to grid-based numerical schemes, the partitioning algorithm gives the closed-form solution without discretization error, and in many cases does not suffer from the curse of dimensionality. The algorithm is applied to two applications. In the main application, we present a model for limit order books with stochastic market depth to study the optimal order execution problem; stochastic market depth is consistent with empirical studies and necessary to accommodate various order activities. The optimal execution policy obtained by the algorithm significantly outperforms that of a deterministic market depth model in numerical examples. In the second application, we use the algorithm to compute the e...