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Practice with Ito Formula

Let \(B_t\)  be a standard Brownian motion. For each of the following definitions of  \(Y_t\), find adapted stochastic process \(\mu_t\) and \(\sigma_t\) so that \(dY_t =\mu_t dt + \sigma_t dB_t\)

  1. \( Y_t =\sin(B_t) \)
  2. \( Y_t= (B_t)^p \) for \(p>0\)
  3. \( Y_t=\exp( B_t – t^2)\)
  4. \(Y_t=\log(B_t) \)
  5. \(Y_t= t^2 B_t \)

Product Chain

Let \(Z_n\) be a collection of independent random variables with \(P(Z_n=1)=\frac12\) and \(P(Z_n=\frac12)=\frac12\) . Define \(X_0=1\) and \(X_{n+1}=Z_n X_n\).

  1. What is \(E( X_n | X_{n-1})\) ?
  2. What is \(E(X_n)\) ?
  3. What is \(\mathrm{Cov}(X_n,X_{n-1})\) ?

Basic Markov Chain I




In each of the graphs pictured, assume that each arrow leaving a vertex has an equal chance of being followed. Hence if there are thee arrows leaving a vertex then there is a 1/3 chance of each being followed.

  1. For each of the six pictures,  find the Markov transition matrix.
  2. State if the Markov chain given by this matrix is irreducible.
  3. If the Matrix is irreducible, state if it is aperiodic.

Uniform distributed points given an arrival

Consider a Poisson arrival process with rate \(\lambda>0\). Let \(T\) be the time of the first arrival starting from time \(t>0\). Let \(N(s,t]\) be the number of arrivals in the time interval \((s,t]\).

Fixing an \(L>0\), define the pdf \(f(t)\) by \(f(t)dt= P(T \in dt | N(0,L]=1)\) for \(t \in (0,L]\). Show that \(f(t)\) is the pdf of a uniform random variable on the interval \([0,L]\) (independent of \(\lambda\) !).

Getting your feet wet numerically

Simulate the following stochastic differential equations:

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

by using the following Euler type numerical approximation

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

where \(n=0,1,2,\dots\) and \(h >0\) is a small number that give the numerical step side.  That is to say that we consider \( X_n \) as an approximation of \(X( t) \) and \( Y_n \) as an approximation of \(Y( t) \) each 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.