If $S_{t}$ satisfies $dS_{t}=\frac{1}{S_{t}}1_{(S_{t}>0)}dB_{t}$, will $S_{t}$ be a martingale?

If the process $S=S_{t}$ satisfies the SDE: $$dS_{t}=\frac{1}{S_{t}}1_{(S_{t}>0)}dB_{t}, \ S_{0}=1.$$ will $S_{t}$ be a martingale? It seems reasonable to say so because $S_{t}$ is clearly nonnegative, and $S_{t}$ will not become unbounded because the term $\frac{1}{S_{t}}$ in the integrand will control the dynamic of $S_{t}$. But I don't know how to formulate a proof. Could anyone help on this? Thank you!

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