We want to solve the PDE $u_t + \left( \frac{x^2 + y^2}{2}\right)u_{xx} + (x-y^2)u_y + ryu = 0 $ where $r$ is some constant and $u(x,y,T) = V(x,y)$ is given. Write an SDE and express $u(x,y,0)$ as the expectation of some function of the path $X_t, Y_t$.
Attempt: I tried to use the multivariate backward equation (2 dimensional) to recover the original SDE's and ended up with $dX_t= \sqrt{x^2 + y^2} dW_t$ and $dY_t = (x-y^2)dt + \sqrt{x^2 + y^2} dW_t$.
The problem I have is recovering the expectation. I'm not too familiar with multidimensional Feynman-Kac, but judging by the $ryu$ term and extrapolating from the one-dimensional case, the desired expectation should have the form E[exp(riemann integral of Y_t)]. Can anyone shed some light on this? Thank you.
EDIT: Oops, wrote the forward equation incorrectly and made a typo, the SDE's have changed