It is known that the Doob-Meyer theorem gives us a unique decomposition,
$N(t)=A(t)+M(t)$
and the compensator part may conditional on a filtration $F_1$: $A(t|F_1)$.
My question is: Does the Doob-Meyer decomposition still exist if we conditional on a different filtration $F_2$? For the moment, I only care about a specific filtration: $F_2 \subset F_1$.
And if so, any example?