  We consider a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$,
which is endowed with two filtrations, $\mathbb{G}$ and $\mathbb{F}$, assumed
to satisfy the usual conditions and such that $\mathbb{F} \subset \mathbb{G}$.
On this probability space we consider a real valued special
$\mathbb{G}$-semimartingale $X$. The purpose of this work is to study the
following two problems:
  A. If $X$ is $\mathbb{F}$-adapted, compute the $\mathbb{F}$-semimartingale
characteristics of $X$ in terms of the $\mathbb{G}$-semimartingale
characteristics of $X$.
  B. If $X$ is not $\mathbb{F}$-adapted, given that the $\mathbb{F}$-optional
projection of $X$ is a special semimartingale, compute the
$\mathbb{F}$-semimartingale characteristics of $\mathbb{F}$-optional projection
of $X$ in terms of the $\mathbb{G}$-canonical decomposition and
$\mathbb{G}$-semimartingale characteristics of $X$.
