Consider the solution $X$ of the stochastic differential equation
$$ \mathrm{d}X_t = \sigma(X_t) \mathrm{d}W_t, $$
where $W$ is a Wiener process and assume the standard growth conditions ($X$ is a Martingale).
Is $X$ subgaussian, e.g., does
$$ \mathbb{E} \exp{sX} \leq \exp \big( \frac {d^2 s^2}{2} \big),\qquad \forall s \in \mathbb{R} $$
hold for some $d>0$?