Currently I am studying finite difference method on derivatives with three (or more) underlyings and little bit confused on operator splitting method because two papers have different result.
For the follwing differential equation, $$ u_\tau(x,y,z,\tau) = rxu_x+ryu_y+rzu_z+\frac{1}{2}\sigma_x^2x^2u_{xx}+\frac{1}{2}\sigma_y^2y^2u_{yy}+\frac{1}{2}\sigma_z^2z^2u_{zz}+\rho_{xy}\sigma_x\sigma_yxyu_{xy}+\rho\sigma_y\sigma_zyzu_{yz}+\rho_{zx}\sigma_z\sigma_xu_{zx}-ru $$
First paper, Comparision of numerical schemes on multi-dimensional black-scholes equations ended up with
$$ \frac{\partial u}{\partial \tau}=L_xu+L_yu+L_zu $$ where $$ L_xu = \frac{1}{2}\sigma_x^2x^2u_{xx}+rxu_x+\frac{1}{2}\rho_{xy}\sigma_x\sigma_yxyu_{xy}+\frac{1}{2}\rho_{xz}\sigma_z\sigma_xu_{xz}-\frac{1}{3}ru \\ L_yu = \frac{1}{2}\sigma_y^2y^2u_{yy}+ryu_y+\frac{1}{2}\rho_{yx}\sigma_y\sigma_xyxu_{yx}+\frac{1}{2}\rho_{yz}\sigma_y\sigma_zu_{yz}-\frac{1}{3}ru \\ L_zu = \frac{1}{2}\sigma_z^2z^2u_{zz}+rzu_z+\frac{1}{2}\rho_{zx}\sigma_z\sigma_xzxu_{zx}+\frac{1}{2}\rho_{zy}\sigma_z\sigma_yu_{zy}-\frac{1}{3}ru $$ (Original paper took derivative wrt $t$ and I changed to $\tau = T-t$ here)
and the second paper, A practical finite difference method for the three-dimensional black-scholes equation defined them as $$ L_xu = \frac{1}{2}\sigma_x^2x^2u_{xx}+rxu_x+\frac{1}{3}\rho_{xy}\sigma_x\sigma_yxyu_{xy}+\frac{1}{3}\rho_{yz}\sigma_y\sigma_zyzu_{yz}+\frac{1}{3}\rho_{xz}\sigma_x\sigma_zxzu_{xz}-\frac{1}{3}ru \\ L_yu = \frac{1}{2}\sigma_y^2y^2u_{yy}+ryu_y+\frac{1}{3}\rho_{yx}\sigma_y\sigma_xyxu_{yx}+\frac{1}{3}\rho_{yz}\sigma_y\sigma_zyzu_{yz}+\frac{1}{3}\rho_{xz}\sigma_x\sigma_zxzu_{xz}-\frac{1}{3}ru \\ L_zu = \frac{1}{2}\sigma_z^2z^2u_{zz}+rzu_z+\frac{1}{3}\rho_{xy}\sigma_x\sigma_yxyu_{xy}+\frac{1}{3}\rho_{zy}\sigma_z\sigma_yzyu_{zy}+\frac{1}{3}\rho_{zx}\sigma_z\sigma_xzxu_{zx}-\frac{1}{3}ru \\ $$
So I wonder which derivation I should use or they are just indifferent
