Endogenous Formation of Limit Order Books: Dynamics Between Trades

In this work, we present a continuous-time large-population game for modeling market microstructure between two consecutive trades. The proposed modeling framework is inspired by our previous work [ Math. Finance, (2017), https://doi.org/10.1111/mafi.12157]. In this framework, the limit order book (LOB) arises as an outcome of an equilibrium between multiple agents who have different beliefs about the future demand for the asset. The agents' beliefs may change according to the information they observe, triggering changes in their behavior. We present an example illustrating how the proposed models can be used to quantify the consequences of changes in relevant information signals. If these signals, themselves, depend on the LOB, then our approach allows one to model the “indirect" market impact (as opposed to the “direct" impact that a market order makes on the LOB, by eliminating certain limit orders). On the mathematical side, we formulate the proposed modeling framework as a continuum-player control-stopping game. We manage to split the equilibrium problem into two parts. The first one is described by a two-dimensional system of reflected backward stochastic differential equations, whose solution components reflect against each other. The second one leads to an infinite-dimensional fixed-point problem for a discontinuous mapping. Both problems are nonstandard, and we prove the existence of their solutions in the paper.