We consider an insider trading equilibrium of Glosten Milgrom type, where an insider maximizes expected profit using his private information, and a market maker sets a fair price according her observation.  To establish an equilibrium, two problems need to be solved simultaneously: insider’s control problem given the price set by the market maker, and market maker’s filtering problem given the insider’s trading strategy. When the risky asset value admits a Bernoulli distribution, we establish the Glosten Milgrom equilibrium and explicitly construct insider’s optimal strategy. This extends results in K. Back and S. Baruch, Econometrica, 72 (2004). When the risky asset value has an arbitrary discrete distribution, we introduce asymptotic Glosten and Milgrom equilibrium and establish its existence. This is based on joint works with Umut Cetin and Cheng Li.