  In this paper, we develop a theory of common decomposition for two correlated
Brownian motions, in which, by using change of time method, the correlated
Brownian motions are represented by a triplet of processes, $(X,Y,T)$, where
$X$ and $Y$ are independent Brownian motions. We show the equivalent conditions
for the triplet being independent. We discuss the connection and difference of
the common decomposition with the local correlation model. Indicated by the
discussion, we propose a new method for constructing correlated Brownian
motions which performs very well in simulation. For applications, we use these
very general results for pricing two-factor financial derivatives whose payoffs
rely very much on the correlations of underlyings. And in addition, with the
help of numerical method, we also make a discussion of the pricing deviation
when substituting a constant correlation model for a general one.
