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mathprobabilitymarkov-chainsrandom-walk

random walk based on previous moves

I am pretty new to this area, so the question that I ask might be straight forward or look naive for other professionals. For a 1D random walk problems, such as drunkard's walk problem, there is no connection between the current move and the previous move, and this problem can be easily solved by absorption Markov Chain method. However, what would happen, if we assume:

(1) the drunkard would have 70% chance walk forward, if previous step was forward, and 30% backward; and

(2) the drunkard would have 30% chance walk forward, if previous step was backward, and 70% backward.

Is there anyway or any recommendation for solving this kind of questions? BTW Monte Carlo is not considered as an excellent option. I would really appreciate the help.

Solution

  • Your state has to contain the last position, so that you have transitions

    (-1,-1) --> (-1,-1)
(+1,+1) --> (+1,+1)

    with 70% probability

    and

    (-1,+1) --> (+1,-1)
(+1,-1) --> (-1,+1)

    with 30% probability each.