Discontinuous Martingales on the interval $[0,T]$

Question

Does there exist a Martingale on continuous time $[0,T]$ such that it is discontinuous for every $t \in [0,T]$?

Answer 1

Yes. Let $\alpha$ be a random variable with uniform distribution on $[0,1]$, we set \begin{align*} X_t=\mathbb{1}_{\alpha+\mathbb{Q}}(t). \end{align*} $X$ is clearly discontinuous everywhere. However, it is a martingale with respect to the filtration defined by ${\cal F}_t=\sigma(\alpha)$ for all $t\geq 0$. Indeed, for all $0\leq s\leq t$, \begin{align*} \mathbb{E}(X_t\mid {\cal F_s})=\mathbb{1}_{\alpha+\mathbb{Q}}(t)=0=X_s\text{ almost surely.} \end{align*}