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I have to prove that a Lévy process is a semimartingale. In general we say that $X$ is a semimartingale if it is an adapted process such that, for each $t ≥ 0$, $$X (t) = X (0) + M(t) + C(t)$$ where $...

Let $(X_t)_{t\ge0}$ be a real-valued Lévy process. Note that $$\mu_t:=\mathcal L(X_t)\;\;\;\text{for }t\ge0$$ is a continuous convolution semigroup$^1$. Let $$\tau_x:\mathbb R\to\mathbb R\;,\;\;\;y\...