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?