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 $\{S_n,n\ge 0\}$ 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 $\{X(t),t\ge 0\}$ that can be represented as the random sum $X(t)=\sum _{i=1}^{N(t)}{Y}_i,t\ge 0$, where $\{N(t),t\ge 0\}$ is a Poisson process and $Y_1,Y_2,\dots$ are iid and independent of $\{N(t),t\ge 0\}$. 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 $E[X(t)]=\lambda tE[Y_1]$ and $\text{Var}(X(t))=\lambda tE[Y_1^2]$.

Q4.3

A nonhomogeneous Poisson process has mean value function

$$m(t)=t^2+2t,t\ge 0$$

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.