formula

$E[\exp(\xi X)|\mathcal{G}]&=&\sum_{j=0}^{\infty}\frac{\xi^j}{j!}E[X^j|\mathcal{G}]\\ &=&\sum_{j=0}^{\infty}\frac{\xi^j}{j!}E[X^j]\quad ({\rm independent})\\ &=&1+\sum_{j=1}^{\infty}\frac{\xi^{2j}}{(2j)!}\frac{(2j)!}{2^jj!}t^{j}\\ &=&1+\sum_{j=1}^{\infty}\frac{1}{j!}\bigg(\frac{\xi^2 t}{2}\bigg)^j\\ &=&\exp\bigg(\frac{1}{2}\xi^2t\bigg)$