This is a continuation of Characterization of martingale diffusions ending in $\{-1,1\}$
$X=(X_t)_{0\le t\le T}$ is said to be a martingle diffusion if $X_0=0$, $X_T\in\{-1,1\}$ and
$$X_t=\int_0^t a(u,X_u)dW_u,\quad \forall t\in [0,T]$$
for some measurable function $a$. What kind of condition on $a$ may ensure the above properties of $X$?
Any answer, comments and references are highly appreciated.
PS : A simple construction is as follows: Take a Brownian motion, denoted by $(B_s)_{s\ge 0}$, in an arbitrary probability space. Set $\tau:=\inf\{s\ge 0: |B_s|\ge 1\}$ and $M= (M_t:=B_{\tau\wedge t})_{t\ge 0}$. Then $M$ is a martingale and $M_{\infty}\in \{-1,1\}$. Taking any (deterministic) time-change, e.g. $f:[0,T]\to [0,\infty]$ with
$$f(t):=\frac{t}{T-t},$$
$X=(X_t:=M_{f(t)})_{0\le t\le T}$ is a required martingale diffusion. Are we able to generalize this construction?
