Right now I am trying to understand how we can use the Fourier theorem in obtaining the formula for option pricing (from Zhu J., "Modular pricing of options").
While modeling the interest rate, he considers two cases: when interest rate and stock returns are correlated and when they are not. In the second case, the stock price has dynamics $dS_t=r(t)S_tdt+vS_tdW_t,$ but while considering the correlation case it is changed to $dS_t=r(t)S_tdt+v\sqrt{r(t)}S_tdW_t.$
I don't understand why is it changed.
I think it is because before we had $r(t)$ only as a drift and we need a interest rate term associated with the Wiener process. Am I right? It still doesn't convince me, so is there any better way to explain it? Also, the changed is made only while modeling the interest rate as a square root model and it doesn't happen while considering interest rate as an Ornstein- Uhlenbeck process. Here, I have no idea why.
