Suposse we have two martingales $N$ and $M$, and the hypothesis of martingale representation theorem holds. Then \begin{equation} N_t=\int_0^t\psi_sdW_s, \,\, M_t=\int_0^t\phi_sdW_s \end{equation} So, my question is there a result that guarantees that is possible to have \begin{equation} N_t=\int_0^t\psi_sdW_s=\int_0^t\frac{\psi_s}{\phi_s}\phi_sdW_s=\int_0^t\frac{\psi_s}{\phi_s}dM_s \end{equation}
For example is possible when $\mathbb{P}(M_t>0)=1$ for all $t\in[0,T]$. Any suggestion or advice is welcome. Thank you!.