Kyle-Back Equilibrium Models and Linear Conditional Mean-field SDEs

Abstract

In this paper we study the Kyle-Back strategic insider trading equilibrium model in which the insider has an instantaneous information on an asset, assumed to follow an Ornstein-Uhlenback-type dynamics that allows possible influence by the market price. Such a model exhibits some further interplay between insider's information and the market price, and it is the first time being put into a rigorous mathematical framework of the recently developed {\it conditional mean-field} stochastic differential equation (CMFSDEs). With the help of the "reference probability measure" concept in filtering theory, we shall first prove a general well-posedness result for a class of linear CMFSDEs, which is new in the literature of both filtering theory and mean-field SDEs, and will be the foundation for the underlying strategic equilibrium model. Assuming some further Gaussian structures of the model, we find a closed form of optimal intensity of trading strategy as well as the dynamic pricing rules. We shall also substantiate the well-posedness of the resulting optimal closed-loop system, whence the existence of Kyle-Back equilibrium. Our result recovers many existing results as special cases.