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 \(I(t,\omega)=\int_0^t \sigma(s,\omega) dB(s,\omega)\).

- 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. - Show that \(M_t\) satisfies the equation

\[ dM_t =\alpha M_t dI_t = \alpha M_t \sigma_t dB_t\]