Random walk with exponential probability - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T04:15:54Z http://mathoverflow.net/feeds/question/88504 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88504/random-walk-with-exponential-probability Random walk with exponential probability Riccardo.Alestra 2012-02-15T10:15:46Z 2012-02-15T13:37:46Z <p>I am trying to solve the following problem:</p> <p>On a segment there are $2N$ points from $-N$ to $N$ passing through zero.</p> <p>A particle jumps from the position $k$ to a position $k+1$ or from a position $k$ to $k-1$ with probability given by $$P(\vert{k}\vert\rightarrow\vert{k+1}\vert)=\frac{1}{2}exp[-\alpha(\vert{N}\vert-\vert{k}\vert)]$$</p> <p>and a probability from $\vert{k}\vert\rightarrow\vert{k-1}\vert$:</p> <p>$$P(\vert{k}\vert\rightarrow\vert{k-1}\vert)=1-\frac{1}{2}exp[-\alpha(\vert{N}\vert-\vert{k}\vert)]$$</p> <p>Starting from a point $k$, the particle begins to jump and the process ends when $\vert{k}\vert>\vert{N}\vert$</p> <p>what is the probability for the process to end after a time $T$ starting from the point $k$?</p> <p>Can someone help me to solve this problem? I posed this problem in a slightly different form also on 'mathematics stackexchange'.</p>