I have to solve the PDE

$$ \begin{align} \frac{\partial F}{\partial t} + \frac{1}{2}\frac{\partial^2 F}{\partial x^2} + \frac{1}{2}\frac{\partial^2 F}{\partial y^2} + \frac{1}{2}\frac{\partial^2 F}{\partial x \partial y} + y &= 0 \\ F(T,x,y) &= e^{x+y} \end{align} $$

but I don't know how to do. Usually, when I have PDEs with cross derivatives, I use a translation $F(t,x,y) = G(t,z)$ (in this case it would be $e^{x+y} = e^z$). But now I have the term "$+y$" that makes this method useless since I get a term "$z-x$", so my PDE is a function both of $z$ and $x$. Do you know how to solve it? Thanks in advance.

  • 1
    $\begingroup$ The Feynman Kac formula applies to parabolic PDEs. Your PDE, however, is elliptic because $\left(\frac{1}{4}\right)^2-\frac{1}{2}\frac{1}{2}<0$. $\endgroup$ – Kevin Sep 12 '20 at 10:53
  • $\begingroup$ So how can I solve that PDE? $\endgroup$ – shot22 Sep 12 '20 at 10:56
  • $\begingroup$ Try some of the standard approaches: reduction to canonical form, separation of variables, Fourier/Laplace transform, Green's function, similarity reduction, generalised functions, ... But that's no question about quantitative finance, it's purely a question about analysis. If you have more boundary conditions, you can set up a finite difference scheme and approximate the solution $\endgroup$ – Kevin Sep 15 '20 at 22:33

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