Decomposition of local martingales

Question

From Kal97, pg. 446:

Theorem 23.14 Any semimartingale $X$ has an a.s. unique decomposition $X=X_0 + X^c + X^d$ where $X^c$ is a continuous local martingale with $X_0^c=0$ and $X^d$ is a purely discontinuous semimartingale.

Q: Is it true that if $X$ is a local martingale then both $X^c$ and $X^d$ are local martingales? If not, does anyone have a counterexample?

My intuition tells me it is true. But I have a very weak handle on this material and cannot find any theorems/corollaries/remarks/etc. in Kal97 or Pro05 that definitively say whether this is the case or not.

Answer 1

Yes; write $X^d = X - X^c - X_0$. Any sum (or difference, or linear combination) of local martingales is again a local martingale.