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

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

X0)=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)=Ut+1/2* sigma^2 Uxx=0

U(x,1)=phi(x_1) 

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