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

On Progressive Filtration Expansions with a Process; Applications to Insider Trading