Semimartingales of Processes with Independent Increments and Enlargement of Filtration

Abstract

Let X be a process with independent increments, $\mathcal{F} = (\mathcal{F}_t )$, $0 \leqq t \leqq T, \mathcal{F} = \sigma (X_s ,s \leqq t)$ a natural filtration. Denote \[ G_t = \sigma \left\{ {X_s ,s \leqq t; X^c \left( T \right); p\left\{ ] {0;T} ]; A \in \mathcal{B} \right\}} \right\},\quad t \leqq T,\] 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.