How can I use the transformations:

$\displaystyle x_1^* = \log(\frac{S_1}{X_1})$ and $\displaystyle x_2^* = \log(\frac{S_2}{X_2})$

and using other transformations analogous to those used in the one-dimensional case.

To show that the two-dimensional Black-Scholes equation can be reduced to the two-dimensional heat equation i.e.

$\displaystyle

\frac{\partial V}{\partial t} + \frac{1}{2}\sum_{i=1}^{2}\sum_{j=1}^{2}\sigma_i\si gma_jS_iS_j\rho_{ij}\frac{\partial^2 V}{\partial^2 S_iS_j} + \sum_{i=1}^{2}(r - D_i) S_i \frac{\partial V}{\partial S_i} - rV = 0

$

can be reduced to the two-dimensional heat equation i.e.

$\displaystyle

\frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2}

$

I am really stuck with it, and don't know how to begin, any help is appreciated.

If any additional details are needed, feel free to ask.