Given a Brownian Motion $B = \lbrace B_t, t \geq 0\rbrace$, Tanaka's Formula provides a decomposition of the submartingale process $|B|$ given by $$|B_t| = \int_0^t sgn(B_s) dB_s +L_0^B(t)$$
where $L_0^B$ is the local time of $B$ at zero.
Are there any other processes which have a similar decomposition? That is, what processes $X$ would satisfy
$$|X_t| = \int_0^t sgn(X_s) dX_s +L_0^X(t)$$
where $L_0^X$ is the local time of $X$ at zero.
I am looking at the proof of Tanaka's formula to see if $B$ can easily be replaced by any martingale or semimartingale. However, as a beginner student in Stochastic Processes, I still cannot fully grasp the proof. However, I am posting this question as a curious student who wants to know the answer to the above question.
Thank you to anyone who can shed light on this.