The following is taken from "Stochastic differential equations and diffusion processes" by Ikeda & Watanabe.
Let $Y,X\in \mathcal M_2^{c,loc}$ (the space of continuous, square integrable, local-martingales).
If we define $\tilde{Y}$ by
$$ \tilde{Y}(t)=Y(t)-\langle Y,X\rangle_t,$$
Then $\tilde{Y}(t)$ is a local martingale.
Proof:
Assume first that $\tilde{Y}(t)$ is bounded in the sense that for each $t\geq 0$, $\tilde{Y}(t)\in \mathfrak L^{\infty}(\Omega)$.
By Ito's formula we have
$$d(M(t)\tilde{Y}(t))=\tilde{Y}(t)dM(t)+M(t)dY(t)$$
where $M(t)$ is the Doléans-Dade exponential of $X$ (we assume that $X$ satisfy the Novikov's condition and hence M(t) is a martingale).
At this point the author say that this implies that $M(t)\tilde{Y}(t)$ is a martingale, shouldn't it be a local-martingale?
We have that $$M(t)\tilde{Y}(t)=\int_0^t \tilde{Y}(t)dM(t)+\int_0^t M(t)dY(t)$$
We have that the first stochastic integral is a martingale since the integrand is bounded and the integrator is a square integrable martingale, now on the second stochastic integral we are integrating wrt a local martingale, I don't see how the martingale property could arise.
Do the author actually mean local-martingale or is there something I am not seeing?