MATH 5050 Miscellaneous Notes


Hints for HW 2

Notes on Uniform integrability

Some guided notes. Fill in the details

Project ideas

Objective: read about something related to probability theory, stochastic proce sses. Then, you can write about something new you learned. Your project can be completely theoretical. You can also read about a topic, and simulate something . It can, obviously, be a combination of the two.

In the end, you will

  1. Make a 15 minute presentation during the last week of class in April
  2. Submit a report in latex by the end of the final exam period.

Some ideas

  1. A simple population model. I was listening to a BBC documentary the other day about mosquitos, and how people are introducing genetically deficient mosquitos into the wild to curb the disease spreading Aedes aegypti mosquito. The genetically deficient mosquitos breed with the normal mosquitos and produce babies that carry the dominant but deficient gene. The deficient mosquitos die out. Can you come up a probabilistic model that models this phenomenon? Perhaps some branching process type model with two trees that interact? Maybe two SDEs that approximates the populations of the mosquitos.

  2. Applications of Brownian motion in Finance. Read “A random walk on wall st” , for example. Understand what an option is, and find out how the Black-Scholes process models option prices. You can simulate the trajectory of a this stocha stic process.

  3. Read about electrical networks and random walks. Read the book by Doyle and Snell. Understand the connection between martingales, harmonic functions and r ecurrence on various graphs.

  4. Read about Markov chains and mixing times. How long does it take for a Mark ov chain to approach its equilibrium distribution? Start with the book by Peres and Levin.

  5. Read more about the Galton-Watson processes (branching processes), continuo us time processes and age dependent processes. Start with the book by Atreya an d Ney.

Guided Quiz 1, Problem 2

If $P$ is an irreducible Markov chain, what is $v(x)$? Assume some general payo ut function $f(x)$ on some state space $S$.

Solution

  1. If $P$ is irreducible, can $P$ have absorbing states? Why or why not?

  2. Let our irreducible chain be labeled by the states $S := {1,2,\ldots,N}$. Using problem 1.7 from Lawler’s book, show that there exist constants $C,\rho$ such that for any two states $j \neq i$

  3. Define the following state mathematically (sans english words): $\bar{j}$ i s the state where the maximum value of $f(x)$ is taken, where $x$ ranges over t he state space $S$.

  4. Mathematically express the probability that you never visit $\bar{j}$ in th e first $n$ steps, and show that it goes to zero.

Guided exercise for uniform integrability

Show that for a positive random variable $X$ such that $\E[ X ] < \infty$

Let’s first establish the integration by parts formula The formula is easy to establish when $X$ has a density $\rho(x)$, and it’s ess entially the same proof for general cdfs $F$ using the Riemann-Stieljes integra l (even though we’ve never carefully defined it). Start by writing down the def inition for

  1. Let $u = x$ and $v = 1 - F(x)$ and apply integration by parts to $u dv$ to get
  2. We need to show that One could apply L’Hospital’s rule here. Can you try it? Under what assumpti ons does it show the above?
  3. Establish the bound
  4. Since is summable, show that for fixed $\e > 0$, there is an $m$ so large that
  5. Conclude that Show that this implies that
  6. Now apply integration by parts again to get Use this identity to prove the result.
  7. Use this result to show that for $\E[ |X| ] < \infty$