4  Generating Random Processes

Q4.1

Suppose that A and B each start with a stake of $10, and bet $1 on consecutive coin flips. The game ends when either one of the players has all of the money. Let \(S_n\) be the fortune of player A at time \(n\). Then \(\left\{ S_n, \; n \ge 0 \right\}\) is a symmetric random walk with absorbing barriers at 0 and 20. Simulate a realization of this process and plot \(S_n\) vs. the time index from time 0 until a barrier is reached.

Q4.2

A compound Poisson process is a stochastic process \(\left\{ X(t), \; t \ge 0 \right\}\) that can be represented as the random sum \(X(t) = \sum_{i=1}^{N(t)} Y_i, \; t \ge 0\), where \(\left\{ N(t), \; t \ge 0 \right\}\) is a Poisson process and \(Y_1, Y_2, \ldots\) are iid and independent of \(\left\{ N(t), \; t \ge 0 \right\}\). Write a program to simulate a compound Poisson(\(\lambda\))-Gamma process (\(Y\) has a Gamma distribution). Estimate the mean and the variance of \(X(10)\) for several choices of the parameters and compare with the theoretical values. Hint: Show that \(E[X(t)] = \lambda t E[Y_1]\) and \(\text{Var}\left(X(t)\right) = \lambda t E[Y_1^2]\).

Q4.3

A nonhomogeneous Poisson process has mean value function

\[ m(t) = t^2 + 2t, \hspace{1em} t \ge 0 \]

Determine the intensity function \(\lambda(t)\) of the process, and write a program to simulate the process on the interval \([4,5]\). Compute the probability distribution of \(N(5)-N(4)\), and compare it to the empirical estimate obtained by replicating the simulation.