  The enlargement of filtration theory is a study of semimartingales when the
basic filtration changes. This theory provides particular techniques on
stochastic calculus. We present here a technique, that we call the local
solution method. We will show, with several examples, that the local solution
method is an effective and flexible method. In particular, with the local
solution method, we will give a unified proof of three of the classical
formulas, namely Jacod's formula, the progressive enlargement formula and the
enlargement formula with honest time. We also show that, besides its
generality, this method, necessiting long (but interesting) computations, may
not be optimal.
