Let $A \subset \mathbb{Z}^2$ be a finite set of points in the square lattice, and let $B$ (the boundary of $A$) be the set of points in $\mathbb{Z}^2 \setminus A$ with at least one (horizontal or vertical) neighbour in $A$. Given any function $g : B \to \mathbb{R}$, construct a function $f : (A \cup B) \to \mathbb{R}$ such that $f\restriction{B} = g$ and, for every $v \in A$, $$f(v) = \frac{1}{4}\sum{_{w~v}} f(w),$$ where the sum if over the $4$ neighbours of $w$.
Let $(X_n)_{n \geq 0}$ be a simple symmetric random walk on $\mathbb{Z}^2$, $X_0 \in A$. Denote $\mathcal{h}_B = \inf \{ n \geq 0 : X_n \in B \}$ the first hitting time of the boundary.
I need to show that $f(X_{\min(n, \mathcal{h}_B)})$, $n \geq 0$ is a martingale relative to the natural filtration of $\mathbb{X}$.
It is clear that the process is adapted, but I am not sure how to proceed with the integrability and the expectation property. I am not sure I understand this set up of the problem either.