We study the optimal execution problem in an order driven market in which the dynamics of the limit order book exhibits a mean-field type behavior among the sellers. Such a problem will contain two underlying sub-problems: one determines the frontier(the best ask price) and the shape of a limit order book (LOB), followed by one that determines the optimal execution strategy. We show that the best ask price is a competitive equilibrium under the so-called Bertrand competition, characterized by a McKean-Vlasov optimal control problem, from which the shape of the LOB will be determined endogenously.
The subsequent optimal execution problem will feature an underlying dynamics as a Reflected McKean-Vlasov SDE with Jumps. We shall prove the well-posedness of such SDE, and validate the Dynamic Programming Principle (DPP). We will then show that the value function is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation which is in the form of a mean-field type integro-partial-differential quasi-variational inequality.
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