Show this equality containing exp using the Itô Lemma

Question

Let $B_t$ be any standard Brownian motion and define $X_t = \text{exp}(B_t)$. I want to prove the following equation: $$dX_t = \text{exp}(B_t)dB_t+\frac{1}{2}\text{exp}(B_t)dt$$ This is supposed to follow directly from Itô's formula, but I don't see how it was used in this case.

Answer 1

Denote $dS_t = dB_t$ and $f(x) = e^x$ then from Ito's lemma it follows that $$dX_t = df(t, S_t) = \left(\frac{\partial f}{\partial t} + \mu_t\frac{\partial f}{\partial x}+\frac{1}{2}\sigma^2_t\frac{\partial^2 f}{\partial x^2}\right)dt + \sigma_t\frac{\partial f}{\partial x}dB_t$$ In our case, $\mu_t = 0, \sigma_t = 1$ hence $$dX_t = \frac{1}{2}e^{B_t}dt + e^{B_t}dB_t$$