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

In: Stochastic Analysis, Filtering, and Stochastic Optimization

Author

Listed:
  • Jin Ma

    (University of Southern California, Department of Mathematics)

  • Eunjung Noh

    (Rutgers University, Department of Mathematics)

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].

Suggested Citation

  • Jin Ma & Eunjung Noh, 2022. "Equilibrium Model of Limit Order Books: A Mean-Field Game View," Springer Books, in: George Yin & Thaleia Zariphopoulou (ed.), Stochastic Analysis, Filtering, and Stochastic Optimization, pages 381-410, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-98519-6_16
    DOI: 10.1007/978-3-030-98519-6_16
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