We study a continuous time equilibrium model of limit order book (LOB) in which the liquidity dynamics follows a non-local, reflected mean-field stochastic differential equation (SDE) with evolving intensity. We will see that the frontier of the LOB (e.g., the best ask price) is the value function of a mean-field stochastic control problem, as the limiting version of a Bertrand-type competition among the liquidity providers.
With a detailed analysis on the N-seller static Bertrand game, we formulate a continuous time limiting mean-field control problem of the representative seller.
We then validate the dynamic programming principle (DPP) and show that the value function is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation.
We argue that the value function can be used to obtain the equilibrium density function of the LOB. (Joint work with Jin Ma)