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March 5th, 2015, 02:09 PM   #1
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some Markov chain questions in Brownian Motion and Birth-Death

Hi,

I need urgent answers. Basically, I don't have background in Markov and I don't need to learn it now actually. But I have to solve the questions below some how. If somebody can give detailed answers to the questions below (From beginning to the final solution with explanations), then I will show a very big appreciation to that person.

1) An urn contains totally N balls. Some balls are black and the other balls are white. A ball is chosen at random at time points whose spacings are independent and exponentially distributed with intensity 3 per minute. The chosen ball is immediately replaced by a ball of the other color. Let Xt denote the number of white balls in the urn at time t.
a) The process {X(t)} is a birth-death process. Find the birth and death rates.
b) Find the stationary distribution. In particular, determine the proportion of time all balls in the urn are white.

2) Give an example of birthrates λ(i) > 0 such that a a birth process {X(t)}t≥0 with birthrates λ(i), i ≥ 0 explodes on average at time t = 17.

3) a) Let Z ∈ N(0, 1) be a standard normal random variable. The process X(t) = sqrt(t)*Z is distributed as a normal random variable at every time, X(0) = 0, and it has continuous trajectories. Is {X(t)} a Brownian motion? (motivate your answer)

b) Suppose {B(t)} and {B˜(t)} are two independent standard Brownian motions and ρ is a constant, −1 < ρ < 1. The process Y(t) = ρB(t) + sqrt(1- ρ^2)*B˜(t) is distributed as a normal random variable at every time, Y0 = 0, and it has continuous trajectories. Is {Y(t)} a Brownian motion? (motivate your answer)
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