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

Question

Quizplus · Accepted Answer

To find $dX_t$, we use Itô's Lemma, which for a function $f(Y_t) = \ln(Y_t)$, gives $df(Y_t) = \frac{\partial f}{\partial Y_t} dY_t + \frac{1}{2} \frac{\partial^2 f}{\partial Y_t^2} (dY_t)^2$. Given $dY_t = b dW_t$, we substitute and simplify to find the correct expression for $dX_t$. None of the other options correctly apply Itô's Lemma to the given functions.

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

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