Solution
= Solution
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
$$
\boxed{b=0\text{ and }K=0}.
$$