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    Suppose we have a process X which either jumps up by 1 or down by 1 each time period, but the probability of an up-jump, given that the process is in state x, is equal to x=(x + 1).
    If h_x denotes the probability of ever hitting 0, given that the starting point is x,

    show that

    h_x -􀀀 h_{x + 1} = (1 -􀀀 h_1)/x!. Hence deduce that h_1 = 1 -􀀀 e􀀀^{1}.

    Please help....


    Do you have the actual question? Do we know anything about the value of x (e.g. is it positive or negative)?

    (Original post by mathsRus)
    You seem to have some typos. Assuming the "up-jump" probability is actually supposed to be x / (x+1):

    Use the same approach you'd use for a normal Gamblers Ruin (or random walk) problem to form a recurrence between h_{x-1}, h_x and h_{x+1}.
    Try to rearrange it to form a recurrence between h_{x+1}-h_x and h_x - h_{x-1}. You can then use this relationship repeatedly to prove the first result (use induction if you really want to be formal).

    For the final result (which I assume is supposed to be h_1 = 1-e^{-1}), assume (or justify, depending on how rigourous you have to be) that \lim_{x\to \infty} h_x = 0. Then use the relationship you proved to find an expression for this limit in terms of h1.
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Updated: October 28, 2015
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