Assume that stochastics process for X follows:
$\frac{dX}{X} = \mu dt + \sigma dW$
I know the answer that close form of X is:
$X = e^{(\mu - \frac{\sigma^2}{2})t + \sigma W(t)}$
However, I dont know what is wrong with the following proof, please help:
$\frac{dX}{X} = d(ln(X)) = \mu dt + \sigma dW$ $
Taking integral on both side:
$ ln(\frac{X}{X_0}) = \mu t + \sigma W(t) $
$ X = X_0 e^{\mu t + \sigma W(t)}$
I am sorry for asking this again. I know that this is a typical mistake but I dont know how to search key word for this.
Thank you