Let (d X_{t}=r X_{t} d t+sigmaleft(X_{t} ight) d W_{t}, Psi) a bounded continuous function and (psi(t, x)=mathbb{E}left(e^{-r(T-t)}

Let \(d X_{t}=r X_{t} d t+\sigma\left(X_{t}\right) d W_{t}, \Psi\) a bounded continuous function and \(\psi(t, x)=\mathbb{E}\left(e^{-r(T-t)} \Psi\left(X_{T}\right) \mid X_{t}=x\right)\). Assuming that \(\psi\) is \(C^{1,2}\), prove that

\[\partial_{t} \psi+r x \partial_{x} \psi+\frac{1}{2} \sigma^{2}(x) \partial_{x x} \psi=r \psi, \psi(T, x)=\Psi(x)\]