Proposition

The space of semimartingales is closed under: 1. Multiplication 2. Absolute change of Measure 3. Composition with $C^{2}$ functions on $\mathbb{R}$ 1. i.e. given $(S_{t})_{t\ge 0}$ semimartingale, $\phi\in C^{2}(\mathbb{R};\mathbb{R})\implies(\phi(S_{t}))_{t\ge 0}$ is a semimartingale