Dynamic equilibrium limit order book model and optimal execution problem

Jin Ma; Xinyang  Wang; Jianfeng  Zhang

doi:10.3934/mcrf.2015.5.557

Article Contents

2015, Volume 5, Issue 3: 557-583. Doi: 10.3934/mcrf.2015.5.557

This issue Previous Article Transposition method for backward stochastic evolution equations revisited, and its application Next Article Generalized homogeneous systems with applications to nonlinear control: A survey

Dynamic equilibrium limit order book model and optimal execution problem

1.
Department of Mathematics, University of Southern California, Los Angeles, CA 90089
2.
Institutional Equity Division, Morgan Stanley, New York, NY 10036

Received: December 31, 2013

Revised: September 30, 2014

Published: August 2015

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Abstract

In this paper we propose a dynamic model of Limit Order Book (LOB). The main feature of our model is that the shape of the LOB is determined endogenously by an expected utility function via a competitive equilibrium argument. Assuming zero resilience, the resulting equilibrium density of the LOB is random, nonlinear, and time inhomogeneous. Consequently, the liquidity cost can be defined dynamically in a natural way.
We next study an optimal execution problem in our model. We verify that the value function satisfies the Dynamic Programming Principle, and is a viscosity solution to the corresponding Hamilton-Jacobi-Bellman equation which is in the form of an integro-partial-differential quasi-variational inequality. We also prove the existence and analyze the structure of the optimal strategy via a verification theorem argument, assuming that the PDE has a classical solution.

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Mathematics Subject Classification: Primary: 91B51, 91B70; Secondary: 93E03, 93E20.

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