Let the domain be $[0,1]^2$. And let $W^x_t$ be the standard Brownian Motion started in $x\in [0,1]^2$ with absorbption on $\partial [0,1]^2$ and choose some $g\in \mathcal{C}^2((0,1)^2)\cap \mathcal{C}_0([0,1]^2)$. Now I would like to extend $g'$ and $g''$ to be continuous on the whole of $[0,1]^2$ and thus hope that I will be able to use Itô's Lemma. So basically I would like for the following to hold true:
$g(W_t)=\int_0^t \nabla g(W_s)d W_s +\int_0^t \Delta g(W_s)ds$
where $\nabla g$ and $\Delta g $ would need to be defined on the Boundary
There are two things troubling me:
- neither $\nabla g$ nor $\Delta g$ are defined on the Boundary (What would then be the best (most sensible) way of extending $g'$ and $g''$?)
- even if one was to define "Derivatives" at the boundary we would still be dealing with a somewhat different Space of Functions and I am not sure whether Itô's Lemma would still hold