 Solution

The semimartingale integration by parts givesHere $d⟨X,M{⟩}_t=h(t)M_{t}dt$. Substitution therefore gives cancellation of the drift:The continuous adapted integrand is locally square integrable. Stopping when $|X|+M$ first exceeds an integer produces true stochastic-integral martingales, hence $Y$ is a local martingale. No unproved global integrability of the product is needed.