We consider the problem of tracking a target whose dynamics is modeled by a continuous Ito semi-martingale. The aim is to minimize both deviation from the target and tracking efforts. We establish the existence of asymptotic lower bounds for this problem, depending on the cost structure. These lower bounds can be related to the time-average control problem of Brownian motion, which is characterized as a deterministic linear programming. A comprehensive list of examples with explicit expressions for the lower bounds is also provided. This is joint work with Jiatu Cai and Peter Tankov.