I'm struggling to solve the following problem for my project:
Given the set of points $(i,j)$ where $i=1,2,3,...,N+1$ and $j=0,1$. Find the probability of getting to the point $(N+1,1)$ from the point $(1,0)$, and from $(2,1)$. Given that the probability to move right is $r$, left is $l$ and up or down is $k$. (so from $j=1$ you can move only down and from $j=0$ only up)
Condition: point $(1,1)$ is absorbing, in other words the probability of getting from $(1,1)$ to $(N+1,1)$ is zero.
I already have some particular solutions for $N=1,2,3,4,..$, but the purpose is to obtain a solution in terms of $r, l, k, N$. I would appreciate any help.
Designate the allowed points in a binary way:
- $0,2,4, \cdots,2N$, for the points $(i,0)$;
- $1,3,5, \cdots,2N+1$, for the points $(i,1)$.
that is
Then we may construct a Markov process, whose transition matrix will be as sketched below.
Only that, from your description, I did not catch exactly what are the movement probabilities at the extreme points.