I was given the following problem:
Let $W$ denote a standard one-dimensional Brownian motion.
Let $S_t = e^{\sigma W_t}$ for $\sigma>0$. Use Ito's lemma to write a stochastic differential equation (SDE) for $dS_t$ in terms of $S_t$.
The solution key starts off as:
$f(x,t) = e^{\sigma x}$. Then $f_x = \sigma f$, and $f_{xx} = \sigma^2 f$. Then $\sigma(x) = 1$...
Here, I'm not able to work out how they got $\sigma(x)=1$. If I have a SDE $dX_t=bdt +\sigma dW_t$ then I know the $\sigma(x)$ function is just the coefficient of $dW_t$ but in this case where I have a solution to a SDE which I have to find I don't understand how to find the $\sigma(x)$ function.
Any help would be much appreciated.