This paper studies an extended continuous-time insider trading model of Calentey and Stacchetti (2010, Econometrica), which allows market makers to observe some partial information about a dynamic risky asset. For each of the two cases with trading until either a fixed time or a random time, we establish the existence and uniqueness of linear Bayesian equilibrium, consisting of insider trading intensity, price pressure on market orders and price pressure on asset observations. It shows that at each of the two equilibria, all information on the risky asset is incorporated in the market price and when the volatility of observation noise keeps constant, the more information observed by market makers, the smaller price pressure on market orders but the greater price pressure on asset observations such that the insider earns less profit and vice versa. It suggests that the partial observation of market makers weakens the information advantage of the insider, which prevents the insider from monopolizing the market to make excessive profit, then reduces the losses of noise traders, thus improving the fairness and effectiveness in the insider trading market.

With the help of the reference probability measure concept in filtering theory, we proved the existence and uniqueness of the ${\mathcal{Q}}^0$-weak solution to a linear conditional mean-field fractional SDE with Hurst parameter $H\in (\frac{1}{2},1)$, which will be a foundation of investigating equilibrium of insider trading driven by fractional Brownian motion.