q-fin updates on arXiv.org
Mon, 02 Mar 2020 06:01:33 GMT language
In this paper 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.
Generalizing the basic idea of Ma et.al. (2015), we argue that the frontier of
the LOB (e.g., the best asking 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, following the idea of Ma et.al. (2015).