Ito's formula for the given expression

Question

$d(g(t)B(t)e^{B(t)})$ how can I calculate this using Ito's formula. I keep getting wrong answers all the time. I am using the Ito's chain rule for $e^{B(t)}$. Obtaining $e^{B(t)}dB(t)+1/2e^{B(t)}dt$.

Answer 1

Let $f(t,B(t)):=g(t)e^{B(t)}$ and then

$$df=g'(t)e^{B(t)}dt+g(t)e^{B(t)}dB(t)+\frac 1 2 e^{B(t)}dt$$

Now you must use the following

$$d( B(t)\cdot f(t,B(t))=B(t)df+fdB(t)+(df)(dB(t)).$$