I have a traditional Gambler's Ruin problem on a set of integers $0,1 \ldots , N$ with transition probabilities of $p(k,k+1) = p(k,k-1) = 1/2$.
In the text book we learned how to compute the expected hitting time in any one of the boundaries ( $0$ or $N$).
Now I face with a little bit different problem: Starting from $0$ what is the expected time to hit the boundary $N$.
My intuition is that I can convert the problem into a new one in which my starting point
is exactly in the middle of a sequence of $2N -1 $ elements and then I can use what we leaned in class regarding the hitting time in any one of the boundaries.
For example, consider a Gambler's Ruin problem on a set of integers $0,1 \ldots , 2N $.
starting from $N$. My hope is that the expected time to hit any of the boundaries ($0$ or $2N$) is the same as my original problem.
Is my intuition correct ?
edit: I start from $0$ and need to get to $N$ and it does not matter if I hit $0$ again and the process will continue in that case.
Best Answer
Assume that $p(0,1)=1$. Then the expected time to hit $N$ from $0$ is exactly $N^2$. To see this, consider a simple random walk $\{S_k\}_{k\ge 0}$, started at $S_0=0$ and stopped when it reaches $\{-N,N\}$. Then your process is simply $\{|S_K|\}_{k \ge 0}$.