formula

$E[\exp(\xi X)|\mathcal{G}]&=&\lim_{n\to\infty}E[\exp(\xi_n x)|\mathcal{G}]\\ &=&\lim_{n\to\infty}\exp\bigg(\frac{1}{2}\xi_n^2t\bigg)\\ &=&\exp\bigg(\frac{1}{2}\xi^2t\bigg)\quad {\rm a.s.}$