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Chris
Posts:
3
Registered:
1/26/08


Difference equation that models random walk
Posted:
Jan 26, 2008 5:45 PM


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 [1p(i)]/2, x(i+1)= x(i)1 with a probability [1p(i)]/2 but x(i) cannot be decreased below 1. The probability p(i) is given by p(i)=1n/[16x(i)]^n*sum_{s=1,..,16x(i)}[16x(i)s]^(n1) 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.



