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

Almost surely differentiable paths must be continuous, so the jump measure must vanish: $K=0$. A nonzero Brownian component has almost surely nowhere-differentiable paths, so also $b=0$. Conversely, if $b=0$ and $K=0$, then $X_t=at$ is differentiable. Thus