Solution
= Solution
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>.