Hi I am trying to calculate E(phi(X(1)) with X(t) satisfies the following

$$d(X(t))=\sigma(X(t))dW(t)$$

$$X(0)=x_0$$

where phi and sigma are arbitrary functions and W(t) is Brownian motion. I think I should apply Feynman–Kac formula but not sure about how to deal with terminal conditions. Is the following right?

$$U(x,t)=U_t+1/2* \sigma^2 U_{xx}=0$$

$$U(x,1)=\phi(x_1) $$

here x_1 is the value of X at time 1. But how do we know about this? We only know x_0...