Application of Fokker-Planck equation in Ito calculus

Question

In http://markov.uc3m.es/2009/02/ito-calculus-for-the-rest-of-us/, is derived. But I don't get this: after all, the process is defined as - which means that $f(X,t)$ in this context is zero (or am I wrong here?). So how can this equation be derived?

Answer 1

Once again, not an ounce of (stochastic-calculus) here, but the remark that $$ \frac{\partial}{\partial x}=2\sqrt{y}\frac{\partial}{\partial y}\implies\frac{\partial^2}{\partial x^2}=2\sqrt{y}\frac{\partial}{\partial y}\left(2\sqrt{y}\frac{\partial}{\partial y}\right). $$ Since $$ \frac{\partial}{\partial y}\left(2\sqrt{y}\frac{\partial}{\partial y}\right)=\frac1{\sqrt{y}}\frac{\partial}{\partial y}+2\sqrt{y}\frac{\partial^2}{\partial y^2}, $$ the result follows.