Some background: define a process $Y_t=\frac{1}{t}\int_0^tX_sds$, where $X_t$ is a standard Brownian motion. Then I define a function $u(t,X_t,Y_t)$ and require it to be a martingale. Thus, by applying Ito formula and setting drift to zero I end up with a following p.d.e.: $$u_t+u_{xx}+\frac{1}{t}(x-y)u_y=0$$$x,y\in \mathbb{R}$, with some terminal condition $u(T,x,y)=f(s,y)$. I approach to solve this problem numerically with finite differences but I have a hard time understanding what is going on at time $0$. At the moment I can't even apply any implicit scheme due to singularity at zero. One can observe though, that due to Lebesgue integration theorem $\lim_{t->0} Y_t=X_0$, however, I divide by $t$, which also approaches zero and thus it is unclear how the pde behaves at that time. Any suggestions how to deal with time zero? This is where I am looking for the solution of this pde. Thanks! 
how to approach time zero when the equation is not defined at that point
pde;finite element;finite difference;parabolic pde