If $$dX_t = \mu(t,X_t)dt + \sigma(X_t)dW_t$$ with $\sigma$ positive, show there exists a function $f$ such that $$d\left(f(X_t)\right) = v(t,X_t)dt + V dW_t$$ where $V$ is constant. How unique is $f$?
The solution by Mark Joshi says: The volatility of $f(X_t)$ will, from Ito's lemma, be $$f'(X_t)\sigma(X_t)$$ So we need to solve $$f'(X_t) = \sigma^{-1}(X_t)A$$ for some constant $A$. We deduce $$f(X) = C + A\int_{0}^{X}\sigma(S)^{-1}dS$$ with $A$ and $C$ arbitrary constants.
Is this a complete solution? I don't see the associated steps in showing if $f$ is unique. Also, I do not understand why when he asserts from Ito's lemma that the volatility part of $f(X_t)$ leads to solving $f'(X_t) = \sigma^{-1}(X_t)A$.