pointwise convergence of semimartingales in probability

Question

In a paper on stochastic finance I'm recently studying, the author defined a closure of some subspace of semimartingales by convergence in probability: $S^N_t\rightarrow S_t$ for each $t$, as $N\rightarrow\infty$, and claimed this convergence defined a metrizable topology. I am not sure how can this be. I do know UCP (uniform convergence on compacts in probability) is metrizable though. Furthermore, I wonder if $S^N$ is RCLL for each N, does the limit process $S$ have an RCLL version? The $S^N$'s are of the form $$ dS_t=S_{t_t}(a_tdt+\sigma_tdW_t+\int_E b(v,t)q(dv,dt))$$ where $W$ is browning motion, and $q$ is a compensated jump measure on a mark space $E\times[0,\infty)$, $a$, $\sigma$ and $b$ are suitably integrable. You can assume further regularities on these integrands if necessary. If these claims aren't always true, what are (some of) the conditions on the integrands such that these claims make sense? Thank you for any suggestions.