  We investigate aspects of semimartingale decompositions, approximation and
the martingale representation for multidimensional correlated Markov processes.
A new interpretation of the dependence among processes is given using the
martingale approach. We show that it is possible to represent, in both
continuous and discrete space, that a multidimensional correlated generalized
diffusion is a linear combination of processes that originate from the
decomposition of the starting multidimensional semimartingale. This result not
only reconciles with the existing theory of diffusion approximations and
decompositions, but defines the general representation of infinitesimal
generators for both multidimensional generalized diffusions and as we will
demonstrate also for the specification of copula density dependence structures.
This new result provides immediate representation of the approximate solution
for correlated stochastic differential equations. We demonstrate desirable
convergence results for the proposed multidimensional semimartingales
decomposition approximations.
