  We study the price-setting problem of market makers under risk neutrality and
perfect competition in continuous time. Thereby we follow the classic
Glosten-Milgrom model that defines bid and ask prices as expectations 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 in a mathematically rigorous way by
solving a 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.
