I have learned the skill to deal with Expected hitting time of one of two barriers.
Now I have a similar one.The walker starts from x=0 and the barriers are located in x=+n.The walker can move one step on the right with probability p and one step on the left with probability q except at point x=0.At x=0,walker move one step on the right with p and stay on the same place with q.Now how to compute $E_n$?
$E_{n}$ represents expected hitting time from x=0 to x=n.$E_{i->j}$represents expected hitting time from x=i to x=j.
I conclude $E_n=E_{n-1}+1+q E_{(n-2->n)}$,$E_{n-2->n}=E_{n}-E_{n-2}$. Then I can get an iteration: $(1-q)E_{n}=E_{n-1}-qE_{n-2}+1$.I can solve it with above skill.
But I can't prove $E_{n-2->n}=E_{n}-E_{n-2}$.Can any one prove it right or wrong?