Validity of Ito's formula for "piecewise-defined" Ito processes Let $b_i: \mathbb{R}^{m} \to \mathbb{R}$ be smooth functions, for each $i \in \{ 1, \ldots,m\}$ and let $(\Omega, \mathbb{F},   \mathbb{P})$ be a probability space equipped with a one-dimensional Brownian motion $W$. We divide the time interval $[0,T]$ evenly into partitions $\{t_r\}_{r=0}^{2N}$ such that $t_r = \frac{rT}{2N}$. On this probability space, we define one-dimensional processes $X^i$ such that they satisfy the following SDE with sign changes in the diffusion:
$$ X^i_t = \int_0^t b_i(X^1_s, \ldots, X^m_s) \,ds + \,W_t, \quad \quad t_0 \leq t \leq t_1, \quad \quad 1 \leq i \leq m.$$
Afterwards, we change the sign of the diffusion for each interval $[t_{i-1}, t_i]$. More precisely,
\begin{eqnarray}
 X^i_t & = & X^i_{t_1} + \int_{t_1}^t b_i(X^1_s, \ldots, X^m_s) \,ds - \,(W_t-W_{t_1}), \quad \quad t_1 \leq t \leq t_2, \quad \quad 1 \leq i \leq m. \\
X^i_t & = & X^i_{t_2} + \int_{t_2}^t b_i(X^1_s, \ldots, X^m_s) \,ds + \,(W_t -W_{t_2}), \quad \quad t_2 \leq t \leq t_3, \quad \quad 1 \leq i \leq m,  \\
& \vdots & \\
X^i_t & = & X^i_{t_{2N-1}} + \int_{t_{2N-1}}^t b_i(X^1_s, \ldots, X^m_s) \,ds - \,(W_t-W_{t_{2N-1}}), \quad \quad t_{2N-1} \leq t \leq t_{2N}, \quad \quad 1 \leq i \leq m.  
\end{eqnarray}

I am wondering the following:
$1. \quad \quad $ Is it true that the quadratic variation $\langle X^i \rangle_t=t,$ for each $t \in [0,T]$? It seems likely, but I am not certain.
$2. \quad \quad $ Is the Ito's formula applicable to processes $X^i$? Again, they seem to be continuous semimartingales, therefore the Ito's formula should be applicable, but I am not entirely sure.


You can write
$$
X_t = \int_0^t b(X_s)ds + \int_0^t \sigma(s) dW_s,
$$
where 
$$
\sigma(t) = \sum_{n=1}^{2N} (-1)^n \mathbf{1}_{[t_{n-1},t_n)}(t).
$$
By the Lévy martingale characterization theorem, the process 
$$
B_t = \int_0^t \sigma(s) dW_s
$$
is a standard Wiener process. Therefore, the answers to your questions are positive.
