# Feynman-Kac representation for a PDE

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!

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 $$dX(t) = rX(t) dt + \sigma X(t) dW(t),\text{ and } dY(t) = h( t,X(t))dt.$$
• In this case, can we write: $$u(x,t,y) = \mathbf{E} [\int_{0}^{T} {h(t,X(t)) dt} \hspace{0.1cm} e^{-r(T-t)} | X(t) = x],$$ – kagami Sep 27 '14 at 8:24