show that the solution is a local martingale iff it has zero drift Most financial maths textbook state the following:
Given an $n$-dimensional Ito-process defined by
\begin{equation}
X_t = X_0 + \int_0^{t} \alpha_s \,d W_s + \int_0^{t} \beta_s \,d s,
\end{equation}
where $(\alpha_t)_{t \geq0}$ is a predictable process that is valued in the space of $n \times d$ matrices and $(W_t)_{t \geq 0}$ is a $d$-dimensional Brownian motion,
\begin{equation}
(X_t) \text{ is a local martingale } \quad \Longleftrightarrow \text{ It has zero drift.}
\end{equation}
Can anyone show me a reference for the proof of this statement or at least give a hint of how to construct this proof? (I know how to prove the ($\Leftarrow$) direction, but I am not so sure about the other one.)

Suppose that $(X_t)_{t \geq 0}$ is a local martingale. Since
$$X_0 + \int_0^t \alpha_s dW_s$$
is also a local martingale, this means that
$$M_t := X_t - \left( X_0 + \int_0^t \alpha_s \, dW_s \right) = \int_0^t \beta_s \, ds$$
is a local martingale. Moreover, $(M_t)_{t \geq 0}$ is of bounded variation and has continuous sample paths. It is widely known that any continuous local martingale of bounded variation is constant, see e.g. Brownian Motion - An Introduction to Stochastic Processes by René Schilling and Lothar Partzsch, Proposition A.22.
