I look for a full characterisation of the continuous martingales $X=(X_t)_{0\leq t\leq T}$ (defined on some filtered probability space as nice as possible) such that
$$X_0=0\quad \mbox{ and } \quad\mathbb P\big[|X_T|=1\big]=1,$$
where $T>0$ is fixed. My thought is as follows : As $X$ is continuous, it must be a time-changed Brownian motion, i.e. there exists some Brownian motion denoted by $W$ (for some different filtration $(\mathcal G_t)$) so that
$$\mathbb P\big[X_t = W_{\langle X\rangle_t},~ \forall 0\leq t\leq T\big] = 1.$$
Set
$$\tau^W := \inf\big\{t>0:~ |W_t|= 1\big\}.$$
Then $\langle X\rangle_T\geq \tau^W$ (which is also a $(\mathcal G_t)-$stopping time). Can this observation help characterise $X$?
Another idea is to consider the non-decreasing functions $f: [0,T]\to [0,\infty]$ with $f(0)=0$ and $f(T)=\infty$. Then the process $(X_t:=W_{f(t)\wedge \tau})_{0\leq t\leq T}$ defines a martingale (with respect to its natural filtration). Does this contraction contains all desired continuous martingales?
