  The Langevin equation with multiplicative noise and state-dependent transport
coefficient has to be always complemented with the proper interpretation rule
of the noise, such as the Ito and Stratonovich conventions. Although the
mathematical relationship between the different rules and how to translate from
one rule to another are well-established, it still remains controversial what
is a more {\it physically} natural rule. In this communication, we derive the
overdamped Langevin equation with multiplicative noise for Brownian particles,
by systematically eliminating the fast degrees of freedom of the underdamped
Langevin equation. The Langevin equations obtained here vary depending on the
choice of the noise conventions but they are different representations for an
identical phenomenon. The results apply to multi-variable, nonequilibrium,
non-stationary systems, and other general settings.

Statistical Mechanics;Soft Condensed Matter