Exponential Martingale

Let \(\sigma(t,\omega)\) be adapted to the filtration generated by a standard Brownian Motion \(B_t\) such that \(|\sigma(x,\omega)| < K\) for some  bound  \(K\) . Let \(I(t,\omega)=\int_0^t \sigma(s,\omega) dB(s,\omega)\).

1. Show that
   \[M_t=\exp\big\{\alpha I(t)-\frac{\alpha^2}{2}\int_0^t \sigma^2(s)ds \big\}\]
   is a martingale. It is called the  exponential martingale. 
2. Show that \(M_t\) satisfies the equation
   \[ dM_t =\alpha M_t dI_t  = \alpha M_t \sigma_t dB_t\]