I have the following partial differential equation, that i am trying to solve,
$$0=A_{t}-A(t,r_{t})\alpha+A_{r}\beta r(t)+A_{rr}\sigma_{r}-c\frac{A_{r}^{2}}{A(t,r_{t})}$$
where $\alpha, \beta$ and c are constants and $A_r$ and $A_t$ are the partial derivatives with respect to $r$ and $t$ respectively, and A(t,r) is the unknown function whose boundary condition is given by,
$A(T,r_T)=1$,
I know that if I don't have the last term in the above equation, I can find the solution using the Feynman kac formula. Any hint or direction will be much appreciated.
Thank you,