Let \(\mathcal F_t\) be a filtration of \(\sigma\)-algebra and \(W_t\) a standard Brownian Motion adapted to the filtration. Define the adapted stochastic process \(X_t\) by
\[ X_t = \alpha_0 \mathbf 1_{[0,\frac12]}(t) + \alpha_{\frac12} \mathbf 1_{(\frac12,1]}(t) \]
where \(\alpha_0\) is a random variable adapted to \(\mathcal F_0\) and \(\alpha_{\frac12}\) is a random variable adapted to \(\mathcal F_{\frac12}\).
Write explicitly the Ito integral
\[\int_0^t X_s dW_s\]
and show by direct calculation that
\[\mathbf E \Big( \int_0^t X_s dW_s\Big) = 0\]
and
\[\mathbf E \Big[\Big( \int_0^t X_s dW_s\Big)^2\Big] = \int_0^t \mathbf E X_s^2 ds\]
