[jason1990]

If you have the time and inclination to write this up, I would interested in reading it. Just to be sure I understand you, you want to model the frog's position as a process indexed by positive and negative time, that is, by a family of random variables {..., S_{-2}, S_{-1}, S_0, S_1, S_2, ...}. You want this process to be Markov in the sense that

E[f(S_n) | S_k for all k < n] = E[f(S_n) | S_{n-1}]

for all integers n. Is this right?

In order for this process to be consistent with the original problem, I would think we would need, for example, to have

P(S_n = k + 2 | S_{n-1} = k) = 1/2

for all n and k. But if the frog is at integer 0 at time 0, then this is not true for n = 0. I would think we would need the position of the frog at time 0 to be random, with a distribution which can be evolved backwards.

To be more specific, if p_n(k) = P(S_n = k), then we want

p_{n+1}(k) = (p_n(k - 2) + p_n(k + 1))/2 for all k.

We seek a family of probability distributions {..., p_{-2}, p_{-1}, p_0, p_1, p_2, ...} that satisfy this recursion. Given p_0, we know how to construct p_n for n > 0. But for some choices of p_0 (e.g., p_0(0) = 1), it will not be possible to construct the distributions p_n for n < 0. Are there any choices of p_0 that work?

Let me know if I am misunderstanding you. I would be interested if you have any links or references to the kind of construction you are envisioning.