Can every càdlàg semi-martingale be written as a sequence of diffusions? That is, is the set of continuous semi-martingales dense in some Skorohod space?



Sequence of diffusions
$C=C[0,1]$, the space of continuous functions on $[0,1]$, is a closed subset in Skorohod space $D=D[0,1]$
(cf. P. Billingsley, *Convergence of Probability Measures*, 2nd ed.(1999) $\S\!\!12$, p.124), hence the set of continuous semi-martingales
**could not be dense** in Skorohod space $D=D[0,1]$.


