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 non-convex stage-wise 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 exact optimal solution to the renewable electricity management problem, for which previously only an approximate solution is known. As a comparison, we show that the approximate solution can be quite inaccurate for some initial states and thus demonstrate an advantage of the exact solution