The Brownian component has finite variance $bt$, and the compensated jump integral has variance
$$
t\int x^2K(dx)
$$
when this integral is finite. Conversely, a finite second moment forces the jump measure to have a finite second moment. Therefore $(X_t^2)$ is integrable exactly when
$$
\boxed{\int_{\mathbb R}x^2K(dx)<\infty}.
$$
These four equivalences are the <Path and moment criteria from a Lévy triplet>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 201 / 6 / d / iv / Solution

{c}

For the convention in the <Lévy–Khintchine formula>, a Lévy process has almost surely differentiable paths exactly when $b=0$ and $K=0$, and continuous paths exactly when $K=0$. It is integrable exactly when $\int_{|x|>1}|x|K(dx)<\infty$, and it has a finite second moment exactly when $\int x^2K(dx)<\infty$.

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