Hi: I will explain my question through the use of an ant that only walks in one direction and it's horizontal and to the right.
So, assume that I have an ant named slowmo who is sitting at $x = 0$. Deterministically, he walks one unit forward to the right per time unit so that, after $N$ time units ( I'll call them periods from now on ), he would be at $x = N$ if he just walked and no wind was involved.
Of course, wind is involved but it's pretty simple wind. After slowmo walks forward one unit to the right at each new discrete time $i$, a random variable from $N(0, \sigma^2)$ is drawn. This random variable, $\epsilon_{i}$, is the amount that slowmo gets pushed forward ( if its +) or backward ( if its -) by the wind after he stepped forward.
Note that, it is possible that slowmo gets pushed forward so frequently that he ends up traveling through the N unit distance in less than N periods. In this case, we still would call his final landing point $x = N$ and his trip is over in that time it took. On the other hand, if he doesn't reach $x=N$ in $N$ periods, then the trip is also considered over and he stops whereever he stopped. In each case, whether he reaches $x = N$ or not, his final location is referred to $X[N]_{\sigma^2}$ and it is a random variable.
So, to summarize above, I put slowmo at $x = 0$, declare the number of time periods he is allowed to walk forward (one step) as $N$, and, after $N$ time periods have passed, the game is finished. His ending location is denoted as $X[N]_{\sigma^2}$.
My question is: What is the variance of $X[N]_{\sigma^2}$ ?
I think the stochastic process itself is a random walk on the integers with drift and an absorption barrier. So, I first tried googling for "random walk on the integers and absorption barrier" but I couldn't find much on it and definitely not the variance of it.
So, I figured it was not worth it to search for the more complex case where drift due to a deterministic function exists also.
This seems like a problem that some past math genius would have solved at some point but like I said, my searches didn't turn up much.
If anyone knows where there is material on this problem, with or without drift, it's appreciated. Clearly, the drift makes it more complicated but, if I found something about the non-drift case, maybe that could lead to more complicated cases like drift or I could atleast get some approximation.
At the same time, although I have no clue, if someone knows how to solve this themself, that would of course be appreciated also. I really don't know how difficult this is but it's beyond my general capabilities for sure. Thanks for the help. Also, if this is not the right stackexchange site to send to, let me know if I should send it to say the math one instead.