On the existence and uniqueness of Glosten-Milgrom price processes

We study the price-setting problem of market makers under perfect competition in continuous time. Thereby we follow the classic Glosten-Milgrom model that defines bid and ask prices as the expectation of a true value of the asset given the market makers partial information that includes the customers trading decisions. The true value is modeled as a Markov process that can be observed by the customers with some noise at Poisson times. We analyze the price-setting problem by solving a non-standard filtering problem with an endogenous filtration that depends on the bid and ask price process quoted by the market maker. Under some conditions we show existence and uniqueness of the price processes. In a different setting we construct a counterexample to uniqueness. Further, we discuss the behavior of the spread by a convergence result and simulations.