Optimal Execution in a Limit Order Book: A Stochastic Control Approach

Abstract

In this dissertation, we study an optimal execution problem under a limit order book (LOB) model. To generalize previous results, we accommodate general utility functions, as well as general order book shapes and volume impact resilience function. By using Dynamic Programming Principle (DPP), the problem is formulated as a singular stochastic control problem. The theory of viscosity solutions of second-order PDEs helps us to identify the value function as the unique solution of a variational inequality, and we are able to numerically calculate it and the corresponding optimal strategy. The generality of the stochastic control approach makes it relatively simple to extend our results to the same problem with an additional budget constraint. Several examples of numerical calculations are presented to show the qualitative behavior of the optimal strategy and how they relate to human intuition.

As an aside, we discovered an innovative way to apply the Dynamic Programming Principle to singular stochastic control problems, and obtained an equation different from the traditional variational inequality. In particular, this new equation leads us to a closed-form solution of the value function in a special case. The arguments leading to the equation are heuristic and we still do not have a general proof to justify the validity of the equation.