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