Consider two complex stochastic processes $(X_t)_{t\ge 0}$ and $(Y_t)_{t\ge 0}$ (adapted to a filtration $(\mathcal{F}_t)_{t\ge 0}$) with the following properties: $$ \begin{align} (1)& \ (X_t) \text{ is continuous.}\\ (2)& \ M_t := \mathrm{e}^{t/2} X_t \text{ is a martingale (w.r.t. $(\mathcal{F}_t)$).}\\ (3)& \ |X_t| \le 1 \; \forall t\ge 0.\\ (4)& \ X_0 = 1. \end{align} $$ The process $(Y_t)$ has the same properties except that $Y_0 = 0$. What I wish to find is something about the long-term behaviour of $|X_t|$ and $|Y_t|$. My conjecture is that $X_t = \mathrm{e}^{\mathrm{i} B_t}$ for a Brownian motion $(B_t)$ and $Y_t = 0 \, \forall t\ge 0$, but I have not been able to deduce any of the two (Actually, knowing that $|X_t| = 1 \forall t \ge 0$ would be enough for my purposes).
One of the corollaries I was able to make is that, since $|M_t|$ and thereby $|M_t|^2$ are sub-martingales, $ \mathbb{E} |X_t|^2 \ge \mathrm{e}^{-t},$ but that is not very interesting for large $t$.
I would very much appreciate any hint at interesting properties that follow from the above or any further conditions that would, in conjunction with the properties above, be sufficient to imply my conjecture.