Apply Ito's Lemma to exponential martingale
$\newcommand{\dd}{\, \mathrm{d}}$
If we apply Ito's lemma,
then
\begin{align*}
\dd \xi_t &amp;= -\xi_t \dd X_t + \frac 12 \xi_t (\dd X_t)^2\\
  &amp;= -\xi_t \left(\frac 12 \lambda_t^2 \dd t + \lambda_t \dd z_t\right) + \frac 12 \xi_t \lambda_t^2 \dd t \\
  &amp;= -\xi_t \lambda \dd z_t.
\end{align*}