Let $M$ be a true real martingale adapted to some brownian motion $B$.
What are the most generic conditions on $M$ to find a deterministic map $\Phi:\mathbb{R}_+\times\mathbb{R} \to \mathbb{R}_+\times \mathbb{R}$ with the following notation $\Phi(s,B_s)=(\Phi_1(s,B_s),\Phi_2(s,B_s))$ and such that $$ M_{\Phi_1(s,B_s)}=\Phi_2(s,B_s) $$
I feel the intuition that when $M$ is solution to nice SDE, it works, but could it be more generic ?
P.S : I know DDS theorem, and it does not answer to this question.