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 denotes the probability of ever hitting 0, given that the starting point is x,
. Hence deduce that .
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A very tough Stochastic Question watch
- Thread Starter
- 25-10-2015 18:48
- 28-10-2015 16:31
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)
- 28-10-2015 17:58
Use the same approach you'd use for a normal Gamblers Ruin (or random walk) problem to form a recurrence between and .
Try to rearrange it to form a recurrence between and . 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 ), assume (or justify, depending on how rigourous you have to be) that . Then use the relationship you proved to find an expression for this limit in terms of h1.