Author Archives: Jonathan Mattingly

Which deck is rigged ?

Two decks of cards are sitting on a table. One deck is a standard deck of 52 cards. The other deck (called the rigged deck)  also has 52 cards but has had 4 of the 13 Harts replaced by Diamonds. (Recall that a standard deck has 4 suits: Diamonds, Harts, Spades, and Clubs. normal there are 13 of each suit.)

  1. What is the probability one chooses 4 cards from the rigged deck and gets exactly 2 diamonds and no hearts?
  2. What is the probability one chooses 4 cards from the standard deck and gets exactly 2 diamonds and no hearts?
  3. You randomly chose one of the decks and draw 4 cards. You obtain exactly 2 diamonds and no hearts.
    1. What is the probability you chose the cards from the rigged deck?
    2. What is the probability you chose the cards from the standard deck?
    3. If you had to guess which deck was used, which would you guess? The standard or the rigged ?

Getting your feet wet numerically

Simulate the following stochastic differential equations:

  • \[ dX(t) = – \lambda X(t) dt + dW(t) \]
  • \[ dX(t) = – \lambda X(t) dt + dW(t) \]

by using the following Euler type numerical approximation

  • \[X_{n+1} = X_n + \lambda X_n h + \sqrt{h} \eta_n\]
  • \[X_{n+1} = X_n + \lambda X_n h + \sqrt{h} X_n\eta_n\]

where \(n=0,1,2,\dots\) and \(h >0\) is a small number which give the numerical step side.  That is to say that we consider \( X_n \) as an approximation of \(X( t) \) with \(t=h n\).  Here \(\eta_n\) are a collection of mutually independent random variables each with a Gaussian distribution with mean zero and variance one. (That is \( N(0,1) \).)

Write code to simulate the two equations using the numerically methods suggested.  Plot some trajectories. Describe how the behavior changes for different choices of \(\lambda\). Can you conjecture where it changes ? Compare and contrast the behavior of the two equations.

Tell your story with pictures.

Handing back tests

A professor randomly hands back test in a class of \(n\) people paying no attention to the names on the paper. Let \(N\) denote the number of people who got the right test. Let \(D\) denote the pairs of people who got each others tests. Let \(T\) denote the number of groups of three who none got the right test but yet among the three of them that have each others tests. Find:

  1. \(\mathbf{E} (N)\)
  2. \(\mathbf{E} (D)\)
  3. \(\mathbf{E} (T)\)

Up by two

Suppose two teams play a series of  games, each producing a winner and a loser, until one time has won two more games than the other. Let \(G\) be the number of games played until this happens. Assuming your favorite team wins each game with probability \(p\), independently of the results of all previous games, find:

  1. \(P(G=n) \) for \(n=2,3,\dots\)
  2. \(\mathbf{E}(G)\)
  3. \(\mathrm{Var}(G)\)

 

 

[Pittman p220, #18]

Population

A population contains \(X_n\) individuals  at time \(n=0,1,2,\dots\) . Suppose that \(X_0\) is distributed as \(\mathrm{Poisson}(\mu)\). Between time \(n\) and \(n+1\) each of the \(X_n\) individuals dies with probability \(p\) independent of the others. The population at time \(n+1\) is comprised of the survivors together with a random number of new immigrants who arrive independently in numbers distributed according to \(\mathrm{Poisson}(\mu)\).

  1. What is the distribution of \(X_n\) ?
  2. What happens to this distribution as \(n \rightarrow \infty\) ? Your answer should depended on \(p\) and \(\mu\). In particular, what is \( \mathbf{E} X_n\) as \(n \rightarrow \infty\) ?

 

 

 

[Pittman [236, #18]

A modified Wright-Fisher Model

 

Consider the ODE

\[ \dot x_t = x_t(1-x_t)\]

and the SDE

\[dX_t = X_t(1-X_t) dt + \sqrt{X_t(1-X_t)} dW_t\]

  1. Argue that \(x_t\) can not leave the interval \([0,1]\) if \( x_0 \in (0,1)\).
  2. What is the behavior of \(x_t\) as \(t \rightarrow\infty\) if if \( x _0\in (0,1)\) ?
  3. Can the diffusion \(X_t\) exit the interval \(  (0,1) \) ? Prove your claims.

No Explosions from Diffusion

Consider the following ODE and SDE:

\[\dot x_t = x^2_t \qquad x_0 >0\]

\[d X_t = X^2_t dt + \sigma |X_t|^\alpha dW_t\qquad X_0 >0\]

where \(\alpha >0\) and \(\sigma >0\).

  1. Show that \(x_t\) blows up in finite time.
  2. Find the values of  \(\sigma\) and \(\alpha\) so that \(X_t\) does not explode (off to infinity).

[ From Klebaner, ex 6.12]

Cox–Ingersoll–Ross model

The following model has SDE has been suggested as a model for interest rates:

\[ dr_t = a(b-r_t)dt +  \sigma \sqrt{r_t} dW_t\]

for \(r_t \in \mathbf R\), \(r_0 >0\) and constants \(a\),\(b\), and \(\sigma\).

  1. Find a closed form expression for \(\mathbf E( r_t)\).
  2. Find a closed form expression  for \(\mathrm{Var}(r_t)\).
  3. Characterize the values of parameters of \(a\), \(b\), and \(\sigma\) such that \(r=0\) is an absorbing point.
  4. What is the nature of the boundary at \(0\) for other values of the parameter ?

 

SDE Example: quadratic geometric BM

Show that the solution \(X_t\) of

\[ dX_t=X_t^2 dt + X_t dB_t\]

where \(X_0=1\) and \(B_t\)  is a standard Brownian motion has the representation

\[ X_t = \exp\Big( \int_0^t X_s ds -\frac12 t + B_t\Big)\]

Practice with Ito and Integration by parts

Define

\[ X_t =X_0 + \int_0^t B_s dB_s\]

where \(B_t\) is a standard Brownian Motion. Show that \(X_t\) can also be written

\[ X_t=X_0 + \frac12 (B^2_t -t)\]