Feynman-Kac No.1

Here is a problem that I found from an old homework assignment.  I've modified the original to make it shorter.  Let W_t be Brownian motion and let 1_z(x) = 1 if x > z and equals 0 otherwise. Consider the function

where ε is a small number.  Write down the PDE satisfied by u_ε(x,t ; a, T).  Show that as ε tends to zero, u_ε(x,t; a, T) approaches the solution of boundary value problem.  Find an analytical expression for u_ε(x,t; a, T).  Use Monte Carlo simulation to evaluate u_ε(0, 0 ; 1/2, 1) where ε = 0.1, 0.001, and 0.0001.

I'll post my attempt at the solution later. Feel free to add a solution.