  In this paper we study progressive filtration expansions with c\`adl\`ag
processes. Using results from the theory of the weak convergence of
$\sigma$-fields, we first establish a semimartingale convergence theorem. Then
we apply it in a filtration expansion with a process setting and provide
sufficient conditions for a semimartingale of the base filtration to remain a
semimartingale in the expanded filtration. Applications to the expansion of a
Brownian filtration are given. The paper concludes with applications to models
of insider trading in financial mathematics.
