Let $(B_t)_{t\geq 0}$ be a standard Brownian motion. Let $\phi: [0,1)\to [0,\infty)$ be defined by $ \phi(t):=t/(1-t)$. Then $(M_t)_{0\le t<1}$ is a continuous Markov martingale with $M_t:=B_{\phi(t)}$. Do we have its martingale representation? Namely, there exist measurable function $\sigma: [0,1)\times \mathbb R\to \mathbb R_+$ and some Brownian motion $W$ (w.r.t. its natural filtration) s.t.

$$M_t=\int_0^t \sigma(s,M_s)dW_t,\quad \forall t\in [0,1).$$