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

Jin Ma () and Eunjung Noh ()
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Jin Ma: University of Southern California, Department of Mathematics
Eunjung Noh: Rutgers University, Department of Mathematics

A chapter in Stochastic Analysis, Filtering, and Stochastic Optimization, 2022, pp 381-410 from Springer

Abstract: Abstract In this paper, we propose a continuous time equilibrium model of the (sellside) limit order book (LOB) in which the liquidity dynamics follows a non-local, reflected mean-field stochastic differential equation (SDE) with state-dependent intensity. To motivate the model we first study an N-seller static mean-field type Bertrand game among the liquidity providers. We shall then formulate the continuous time model as the limiting mean-field dynamics of the representative seller, and argue that the frontier of the LOB (e.g., the best ask price) is the value function of a mean-field stochastic control problem by the representative seller. Using a dynamic programming approach, we show that the value function is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation, which can be used to determine the equilibrium density function of the LOB, in the spirit of [32].

Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-98519-6_16

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DOI: 10.1007/978-3-030-98519-6_16

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