Some questions about Brownian Motion and Birth-Death in Markov chain

[power3173]

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 somehow. 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)

 

[chisigma]

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 somehow. 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)

Let's start with step 1) ... limiting ourselves to the population of white balls, calling $\lambda_{n}$ the birth probability and $\mu_{n}$ the death probability for each state with n=0,1,...,N You have...

$\displaystyle \lambda_{n} = \frac{N - n}{N}$

$\mu_{n} = \frac{n}{N}\ (1)$

Calling $p_{n} (t)$ the probability to be in the state n at the time t, the steady state value of the $p_{n}$ are given by...

$\displaystyle p_{0} = (1 + \sum_{n=1}^{N} \frac{\lambda_{0}\ \lambda_{1}\ ...\ \lambda_{n-1}}{\mu{1}\ \mu_{2}\ ...\ \mu_{n}})^{-1} = (\sum_{n = 0}^{N} \binom{N}{n} )^{-1} = 2^{- N}\ (2)$

$\displaystyle p_{n} = \frac{\lambda_{0}\ \lambda_{1}\ ...\ \lambda_{n-1}}{\mu{1}\ \mu_{2}\ ...\ \mu_{n}}\ p_{0} = \binom {N}{n}\ 2^{- N}\ (3)$

... so that the distribution is binomial...

Kind regards

$\chi$ $\sigma$

 

[chisigma]

Question 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)

A Brownian motion is a stochastic process $W_{t}$ with p.d.f. ...

$\displaystyle f_{W} (x,t) = \frac{1}{\sqrt{2\ \pi\ t}}\ e^{- \frac{x^{2}}{2\ t}}\ (1)$

... i.e. with mean $\mu = 0$ and $\sigma^{2} = t$ ...

If Z is a standard r.v. with mean $\mu = 0$ and $\sigma^{2}=1$ and $X = a(t)\ Z$ is a stochastic, then X has mean $\mu=0$ and $\sigma^{2} = a^{2} (t)$, so that if $a(t)=\sqrt{t}$, then X is a Brownian motion...

Kind regards

$\chi$ $\sigma$

 

[chisigma]

Question 3 b): Suppose {A(t)} and {B(t)} are two independent standard Brownian motions and ρ is a constant, −1 < ρ < 1. The process Y(t) = ρ A(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)

If A(t) and B(t) are both Brownian and independent, then $\rho\ A(t)$ has variance $\rho^{2}\ t$ and $\sqrt{1 - \rho^{2}}\ B(t)$ has variance $(1 - \rho^{2})\ t$, so that Y(t) has variance $(\rho^{2} + 1 - \rho^{2})\ t = t$ and it is also Brownian...

Kind regards

$\chi$ $\sigma$

 

[power3173]

Let's start with step 1) ... limiting ourselves to the population of white balls, calling $\lambda_{n}$ the birth probability and $\mu_{n}$ the death probability for each state with n=0,1,...,N You have...

$\displaystyle \lambda_{n} = \frac{N - n}{N}$

$\mu_{n} = \frac{n}{N}\ (1)$

Calling $p_{n} (t)$ the probability to be in the state n at the time t, the steady state value of the $p_{n}$ are given by...

$\displaystyle p_{0} = (1 + \sum_{n=1}^{N} \frac{\lambda_{0}\ \lambda_{1}\ ...\ \lambda_{n-1}}{\mu{1}\ \mu_{2}\ ...\ \mu_{n}})^{-1} = (\sum_{n = 0}^{N} \binom{N}{n} )^{-1} = 2^{- N}\ (2)$

$\displaystyle p_{n} = \frac{\lambda_{0}\ \lambda_{1}\ ...\ \lambda_{n-1}}{\mu{1}\ \mu_{2}\ ...\ \mu_{n}}\ p_{0} = \binom {N}{n}\ 2^{- N}\ (3)$

... so that the distribution is binomial...

Firstly, again thanks a lot for your answers.

Just some points I didn't understant completely.

I think P(0) and P(n) are the stationary distributions, right? Then what can we say about birth and death rates?

And what about the question 1.b, the proportion of time all balls in the urn are white?

Thanks...

 

[chisigma]

Firstly, again thanks a lot for your answers.

Just some points I didn't understant completely.

I think P(0) and P(n) are the stationary distributions, right? Then what can we say about birth and death rates?

And what about the question 1.b, the proportion of time all balls in the urn are white?

Thanks...

The birth and death rate in the unit time [in this case 20 seconds...] are given by...

$\displaystyle R_{b} = R_{d} = \sum_{n=0}^{N} \lambda_{n}\ p_{n} = \sum_{n=0}^{N} \mu_{n}\ p_{n} = 2^{- N}\ \sum_{n=0}^{n} \binom {N-1}{n-1} = 2^{- N}\ 2^{N-1} = \frac{1}{2}\ (1)$

The proportion of time in which alla balls are white is...

$\displaystyle p_{N} = 2^{- N}\ (2)$

Kind regards

$\chi$ $\sigma$