Dynamic equilibrium model for limit order book and optimal execution problem

In this dissertation we study the optimal execution problem on an order driven market under our equilibrium model for the limit order book (LOB). In our model the dynamics of the price of a financial security can be decomposed into two independent elements, the fundamental price and the supply / demand of the market. The main feature of our model is that the shape of the LOB can be determined endogenously by an expected return function via a competitive equilibrium argument. The resulting equilibrium distribution is random, nonlinear, and time inhomogeneous. Thus the liquidity cost and the price impact of large trades are self-contained in our model, and can be dynamically defined in a natural way. Assuming the zero resilience, we argue that the liquidity cost should be essentially linear with respect to the instantaneous trade size. As a consequence, we show that the optimization problem could be formulated as a special finite-fuel type of singular stochastic control problem. We verify the Dynamic Programming Principle (DPP) in this case, and prove that the value function is a viscosity solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation, which is in the form of a quasi-integro-variational inequality (QIVI) as expected. We then construct the optimal portfolio strategy using the classical solution to the HJB equation, and it turns out to contain both singular and regular continuous parts. We conclude the work and make remarks on some topics for future research at the end.