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Topic: Difference equation that models random walk
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Posts: 3
Registered: 1/26/08
Difference equation that models random walk
Posted: Jan 26, 2008 5:45 PM
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Let x(i) be a discrete Markov process (random walk) where:
x(i+1)= x(i)+1 with a probability p(i),
x(i+1)= x(i) with a probability [1-p(i)]/2,
x(i+1)= x(i)-1 with a probability [1-p(i)]/2 but x(i) cannot be decreased below 1.
The probability p(i) is given by
where n is an integer constant number.
As follows from the corresponding formula: 0<= p(i)<1 and p(i) is a strictly decreasing function of i.

Please, help me to formulate the stochastic difference equation that models this random walk.

I expect that the difference equation has an equilibrium for p(i)=1/3, which is independent of n. The latter may be computed according to the condition that the expected change of x(i) at the equilibrium equals zero.

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