Consider the case where I have a discrete random walk on the integers over an interval $[0, M]$, where I start at some position $0 < k < M$, and both endpoints (i.e. $0$ and $M$) are fully absorbing. The caveat, to this otherwise simple random walk problem, is that the probability of taking a step depends on the distance between the walker and the absorbing boundaries. Letting $x_i$ represent the position of the walker, we have $P[+1] = \frac{x_i}{M}$, and $P[-1] = 1 - P[+1]$.
In other words, the probability of the walker taking a $+1$ step is equal to the probability of sampling from the integers over the interval $[1, M]$, with uniform probability, and seleting an integer $j$ such that $j \leq x_i$.
Can we state the probability that the walker reaches the absorbing target $M$?