Equilibrium model of limit order book and optimal execution problem

2018-08-10In this dissertation, we study an equilibrium model of a limit order book (LOB) and an optimal execution problem. To generalize the previous results, we accommodate the idea of Bertrand price competition, as well as nonlocal mean-field stochastic differential equation (SDE) with evolving intensity and reflecting boundary conditions. To describe the equilibrium of LOB, we start with N sellers’ static Bertrand game, and extend the model to continuous time setting to formulate it as a mean-field type control problem of the representative seller, who wants to maximize the discounted lifelong expected utility. Using dynamic programming principle (DPP), we could form a Hamilton-Jacobi-Bellman (HJB) equation and prove the value function is the viscosity solution of it. And, we show the value function can be used to obtain the equilibrium density function of a LOB. ❧ Assuming the LOB has reached at the equilibrium, we solve the optimal execution problem of a buyer, who wants to purchase a certain number of shares during the finite time horizon with minimum cost. Again, we show that the Bellman principle of DPP holds in this case, and the value function is the viscosity solution of the HJB quasi-variational inequality (QVI). Also, we investigate the optimal strategy when the QVI has a classical solution