I am trying to solve the following problem:
On a segment there are $2N$ points from $-N$ to $N$ passing through zero.
A particle jumps from the position $k$ to a position $k+1$ or from a position $k$ to $k-1$ with probability given by $$P(\vert{k}\vert\rightarrow\vert{k+1}\vert)=\frac{1}{2}exp[-\alpha(\vert{N}\vert-\vert{k}\vert)]$$
and a probability from $\vert{k}\vert\rightarrow\vert{k-1}\vert$:
$$P(\vert{k}\vert\rightarrow\vert{k+1}\vert)=1-\frac{1}{2}exp[-\alpha(\vert{N}\vert-\vert{k}\vert)]$$$P(\vert{k}\vert\rightarrow\vert{k-1}\vert)=1-\frac{1}{2}exp[-\alpha(\vert{N}\vert-\vert{k}\vert)]$$
Starting from a point $k$, the particle begins to jump and the process ends when $\vert{k}\vert>\vert{N}\vert$
what is the probability for the process to end after a time $T$ starting from the point $k$?
Can someone help me to solve this problem? I posed this problem in a slightly different form also on 'mathematics stackexchange'.
