Point process bridges and weak convergence of insider trading models

TL;DR

This paper constructs a specific stochastic process to demonstrate the existence of equilibrium in insider trading models and proves the weak convergence of these equilibria to Kyle-Back models as trading intensity increases.

Contribution

It explicitly constructs a bridge process for Poisson differences, establishing the existence of Glosten-Milgrom equilibrium and its convergence to Kyle-Back equilibrium without extra assumptions.

Findings

Constructed a bridge process with specified distribution properties.

Proved existence of Glosten-Milgrom equilibrium with mixed insider strategies.

Showed weak convergence of equilibria to Kyle-Back model as intensity grows.

Abstract

We construct explicitly a bridge process whose distribution, in its own filtration, is the same as the difference of two independent Poisson processes with the same intensity and its time 1 value satisfies a specific constraint. This construction allows us to show the existence of Glosten-Milgrom equilibrium and its associated optimal trading strategy for the insider. In the equilibrium the insider employs a mixed strategy to randomly submit two types of orders: one type trades in the same direction as noise trades while the other cancels some of the noise trades by submitting opposite orders when noise trades arrive. The construction also allows us to prove that Glosten-Milgrom equilibria converge weakly to Kyle-Back equilibrium, without the additional assumptions imposed in \textit{K. Back and S. Baruch, Econometrica, 72 (2004), pp. 433-465}, when the common intensity of the Poisson processes tends to infinity.