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

The Brownian and drift components are continuous. Any nonzero Lévy measure produces jumps: a set bounded away from zero with positive finite $K$-measure gives a nontrivial compound Poisson component, and increasing such sets detects every nonzero $K$. Hence paths are almost surely continuous exactly when