How can we check whether a parabolic PDE can be transformed into a heat equation?

By Transformation from the Black-Scholes differential equation to the diffusion equation - and back, we are able to transform the PDE $$\frac{\partial V}{\partial t} +\frac{1}{2}\sigma^2S^2\frac{\partial ^2 V}{\partial S^2} +rS\frac{\partial V}{\partial S} - rV=0$$ into a heat equation.

After I turn this equation into 2D by adding a term $$S\frac{\partial V}{\partial J}$$, we have $$\frac{\partial V}{\partial t} +\frac{1}{2}\sigma^2S^2\frac{\partial ^2 V}{\partial S^2} +rS\frac{\partial V}{\partial S} + S\frac{\partial V}{\partial J}- rV=0$$

Is there an algorithm that helps us transform this PDE into a heat equation, or at least eliminate the $$S$$ in the coefficient? (Since $$S$$ is a variable while the other coefficients are constant)

• @Mattos No, we don't' have $V_{JJ}$ in the pde. Any thoughts? Jul 1, 2019 at 1:53
• @Mattos Exactly... Jul 1, 2019 at 1:56
• @Mattos It’s Black-Scholes PDE for Asian options. I’m sure the form is correct. So you mean that it’s nontrivial to transform it into a heat equation? Jul 1, 2019 at 1:58
• The first link on the google search yields this. Jul 1, 2019 at 2:01
• @Mattos thanks a lot Jul 1, 2019 at 2:08

Hint:

Let $$V=e^{rt}W$$ ,

Then $$\dfrac{\partial V}{\partial t}=e^{rt}\dfrac{\partial W}{\partial t}+re^{rt}W$$

$$\dfrac{\partial V}{\partial S}=e^{rt}\dfrac{\partial W}{\partial S}$$

$$\dfrac{\partial^2V}{\partial S^2}=e^{rt}\dfrac{\partial^2W}{\partial S^2}$$

$$\dfrac{\partial V}{\partial J}=e^{rt}\dfrac{\partial W}{\partial J}$$

$$\therefore e^{rt}\dfrac{\partial W}{\partial t}+re^{rt}W+\dfrac{\sigma^2S^2}{2}e^{rt}\dfrac{\partial^2W}{\partial S^2}+rSe^{rt}\dfrac{\partial W}{\partial S}+Se^{rt}\dfrac{\partial W}{\partial J}-re^{rt}W=0$$

$$\dfrac{\partial W}{\partial t}+\dfrac{\sigma^2S^2}{2}\dfrac{\partial^2W}{\partial S^2}+rS\dfrac{\partial W}{\partial S}+S\dfrac{\partial W}{\partial J}=0$$