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In this paper we study a continuous time equilibrium model of limit orderbook (LOB) in which the liquidity dynamics follows a non-local, reflectedmean-field stochastic differential equation (SDE) with evolving intensity.Generalizing the basic idea of Ma et al. (2015), we argue that the frontier ofthe LOB (e.g., the best asking price) is the value function of a mean-fieldstochastic control problem, as the limiting version of a Bertrand-typecompetition among the liquidity providers. With a detailed analysis on the$N$-seller static Bertrand game, we formulate a continuous time limitingmean-field control problem of the representative seller. We then validate thedynamic programming principle (DPP), and show that the value function is aviscosity solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation.We argue that the value function can be used to obtain the equilibrium densityfunction of the LOB, following the idea of Ma et al. (2015).

Equilibrium Model of Limit Order Books: A Mean-field Game View