In the previous post, we mentioned the Skorokhod embedding theorem. This theorem gives the probability for the stopping time T as one of the probabilities in the distribution.  In this post, we try to prove it.  We define a pair of random variables (A,B) that takes values anywhere in the N x N grid with the following distribution:
the initial state with A=0 and B=0 to have a probability p0 and the
any other state with A=i and B=j to have a probability as a normalization factor times j-i times the combination of the independent probabilities for i and j from the distribution.
We can now cumulate all the other states to have a probability of 1 - p0. We can split the (j-1) to two separate terms.
Using the zero mean equation in the cumulation equation, we can apply the above two for all i < 0 < j, we get that the normalization factor is the inverse of this common value : sum of i and pi over all i >  0
Assuming the stopping time as an infinite series of Sn = i and (A,B) to be independent of Sn, we can take a new random variable Z in terms of the intersection of the random walk to reach value TA and before it reaches TB.
The claim is that the probability for this random variable to take a value k is the corresponding probability from the distribution.  For k = 0,  Z = 0 has a probability p0. For k > 0 we can use the theorem that computes the probability of the random variable we defined and we see that this has a value pk.