Given that $d Y _ { t } = b d W _ { t }$ and $X _ { t } = e ^ { Y t }$ ,what is $d X _ { t }$ ?
A) $e ^ { b d W _ { t } }$
B) $X _ { t } e ^ { b ^ { 2 } dt + Y _ { t } d W _ { t } }$
C) $\ln \left( X _ { t } \right) \left[ \frac { 1 } { 2 } b ^ { 2 } d t + b d W _ { t } \right]$
D) $b X _ { t } \left[ \frac { 1 } { 2 } b d t + d W _ { t } \right]$

Question

Quizplus · Accepted Answer

Using Itô's lemma, $dX_t = X_t \left( \frac{1}{2} b^2 dt + b dW_t \right)$.

Given That dYt=bdWtd Y _ { t } = b d W _ { t }dYt​=bdWt​

Given That $d Y _ { t } = b d W _ { t }$