I was going through this paper http://www.math.ku.dk/~rolf/teaching/ctff03/Heston93.pdf. I am stuck at page number 341 equation A2.
Given, $$dx(t)=(r+u_j\nu)dt + \sqrt{\nu(t)}dz_1(t)$$ $$d\nu = (a_j-b_j\nu)dt+\sigma \sqrt{\nu(t)}dz_2(t)$$ $$f(x,\nu, t)=E[g(x(T), \nu(T))|x(t)=x, \nu(t)=\nu]$$
Using Ito's lemma, I am getting
$$df = \frac{\partial f}{\partial t}dt+\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial \nu}d\nu + \frac{1}{2}\nu(t)\frac{\partial ^2f}{\partial x^2}dt + \frac{1}{2}\sigma ^2 \nu(t)\frac{\partial ^2f}{\partial \nu^2}dt+\rho \sigma \nu \frac{\partial ^2f}{\partial \nu \partial x}dt $$
$$=(\frac{\partial f}{\partial t}+(r+u_j\nu)\frac{\partial f}{\partial x}+(a_j-b_j\nu)\frac{\partial f}{\partial \nu}+\frac{1}{2}\nu\frac{\partial ^2f}{\partial x^2}+\frac{1}{2}\sigma^2\nu\frac{\partial ^2f}{\partial \nu^2}+\rho\sigma\nu\frac{\partial ^2f}{\partial \nu \partial x})dt$$ $$+\sqrt{\nu(t)}\frac{\partial f}{\partial x}dz_1+\sigma\sqrt{\nu(t)}\frac{\partial f}{\partial \nu}dz_2$$
But equation A2 in Appendix is given as
$$df=(\frac{\partial f}{\partial t}+(r+u_j\nu)\frac{\partial f}{\partial x}+(a_j-b_j\nu)\frac{\partial f}{\partial \nu}+\frac{1}{2}\nu\frac{\partial ^2f}{\partial x^2}+\frac{1}{2}\sigma^2\nu\frac{\partial ^2f}{\partial \nu^2}+\rho\sigma\nu\frac{\partial ^2f}{\partial \nu \partial x})dt$$ $$+(r+u_j\nu)\frac{\partial f}{\partial x}dz_1+(a-b_j\nu)\frac{\partial f}{\partial \nu}dz_2$$
So, for the last two terms I am getting something different.
Instead of $(r+u_j\nu)$, I am getting $\sqrt{\nu}$ and instead of $(a-b_j\nu)$, I am getting $\sigma \sqrt{\nu}$.
Please let me know why am I getting different expressions ??