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?
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1$\begingroup$ @JGWang settles the matter when $D$ carries the Skorohod J1 topology. But have you thought about employing the M1 topology? It's not hard to see, for example, that the path $$ x_n(t) =\cases{0, &$0\le t\le 1/2-1/n$, \cr n(t-1/2+1/n), &$1/2-1/n\le t\le 1/2$,\cr 1.&$1/2\le t\le 1$,\cr} $$ converges to the path $x(t)=1_{[1/2,1]}(t)$ in the M1 topology. $\endgroup$– John DawkinsCommented Jun 11, 2017 at 16:44
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$\begingroup$ In this case, under one of the other Skorohod topologies would the claim be true? $\endgroup$– ABIMCommented Jun 12, 2017 at 4:21
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1 Answer
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$C=C[0,1]$, 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 $D=D[0,1]$.
