Feynman-Kac representation for a PDE

Question

I have the following PDE: $$ u_t + r x u_x + \frac{\sigma^2 x^2}{2} u_{xx} + h(t,x) u_y - ru =0 $$ $$ u(x,T,y) = y $$

I wanted to check whether the following representation is correct (I used Feynman-Kac theorem):

$$ u(x,t,y) = E[ y e^{-r(T-t)} | x(T) = x] $$

Thanks!

Answer 1

What I get is $$ u(x,t,y) = \mathbf{E} \left[ y(T)e^{-t(T-t)} \right. \left| X(t) = x, Y(t) = y \right], $$ where the processes $X$, and $Y$ follow $$ dX(t) = rX(t) dt + \sigma X(t) dW(t),\text{ and } dY(t) = h( t,X(t))dt. $$