Since the probability is always the same regardless of past state, you don't need to use martingale or markov models.
Just use a standard 2D random walk in discrete time and space.
If I had an integer lattice on a plane and a particle on a point on the lattice in the first quadrant, what is a good way to model the probability of the particle reaching either the x or y axis? The probability of the particle going up, down, left, right is always 1/4. And the particle always moves on each step.
I'm assuming I need to consider Markov chains and martingales, but any advice would be helpful!
Chiro, but don't you use martingales exactly when future states aren't affected by anything but the last state?
Also, what is the standard 2D random walk formulation that you are talking about? Perhaps there is a literature you can refer me to?
You are right about the martingale aspect.
The 2D random walk looks like this:
Let Z(n) be the position at time n point in the form (x(n),y(n)).
Then Z(n+1) = Z(n) + D
where D takes on four values with equal probability (1,0), (0,1), (0,-1), (-1,0)
Z(n+x) is independent from Z(n) where x is non-zero and valid.
You can use a tree argument or an analytic argument to derive the probability of landing in a particular cell after n moves.
I would start by looking at the tree argument and enumerated possible moves after p time steps and then use that to further investigate the probability distribution for Z(n) = (x,y).