Let $I(\cdot)$ be an indicator, and $B_{t}$ be an 1-dim standard Brownian motion in a nice filtered probability space $(\Omega, \mathcal{F}, P, \mathcal{F}_{t})$. We consider a random process $$Y_{t} = I(t\ge 1) B_{t-1}.$$ Obviously, $Y$ is an $\mathcal F_{t}$-adapted process with finite quadratic variation $\langle Y \rangle_{t} = (t-1) I(t\ge 1)$.

[Q1.] Is it a Ito process, i.e. there exists a representation of the form $$Y_{t} = \int_{0}^{t} b_{s}ds + \sigma_{s}d B_{s}, \quad \forall t\ge 0$$ for some $\mathcal F_t$-adapted processes $b$ and $\sigma$?

[Q2.] If No for [Q1.], then is it semi-martingale?

[Q3.] If the answers for the above are both NO, then it gives an example of finite QV which is not semi-martingale. Is there any other such an example which belongs to QV but not in semi-martingale?