$dZ_t = Z_t X_t \, dW_t$ is this Processes a martingale?

Question

In the exercise 2.28 of Karatzas and Shreve - Brownian Motion and Stochastic Calculus one reads (pg 147):

But in the sequence, after we have established Itô's calculus, one reads (pg 153)

It seems to me that if $dZ_t = Z_t X_t \, dW_t$ then $Z_t $ is a local martingale. If in addition we have that $Z$ is a supermartingale, then we have that $Z_t$ is a martingale.

The questions therefore are:

Is $Z$ a martingale?

Why do we say in problem 2.28 that the process $Z$ is a supermartingale and in the sequel no mention is made to the case that $Z$ is a martingale?

Answer 1

In this case, $Z$ is always a non-negative local martingale, and hence also a supermartingale (by Fatou's lemma). It need not be a martingale, however.