Geodesic
Semimartingales of processes with independent increments and enlargement of filtration
Teoriâ veroâtnostej i ee primeneniâ, Tome 38 (1993) no. 3, pp. 491-502

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Let $X$ be a process with independent increments, $\mathcal{F} = (\mathcal{F}_t )$, $0 \le t \le T, \mathcal{F} = \sigma (X_s ,s \le t)$ a natural filtration. Denote $$ G_t = \sigma \{ {X_s ,s \le t; X^c ( T ); p\{ ] {0;T} ]; A \in \mathcal{B} \}} \},\qquad t \le T, $$ where ${X^c }$ is a continuous martingale component, ${p\{ { ] {0;T} ]; A \in \mathcal{B}}\}}$ is the integer-valued Poisson measure generated by ${X,\mathcal{B}}$ is the Borel $\sigma $-algebra. The paper discusses conditions under which any process $Y$ being a semimartingale with respect to filtration $F$ is also a semimartingale with respect to filtration $G$.
Keywords: processes with independent increments, semimartingales, extension of a filtration flow. 
@article{TVP_1993_38_3_a1,
     author = {L. I. Gal'chuk},
     title = {Semimartingales of processes with independent increments and enlargement of filtration},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {491--502},
     publisher = {mathdoc},
     volume = {38},
     number = {3},
     year = {1993},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1993_38_3_a1/}
}
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L. I. Gal'chuk. Semimartingales of processes with independent increments and enlargement of filtration. Teoriâ veroâtnostej i ee primeneniâ, Tome 38 (1993) no. 3, pp. 491-502. http://geodesic.mathdoc.fr/item/TVP_1993_38_3_a1/