R exercise #6

This is a function that plots 10 replicates of a Brownian Motion (if the
parameter  a  is  0)  or 10 replicates of an Ornstein-Uhlenbeck process
if  a  is set to a value greater than  0.   Try values of  a  from about
0.001  to  0.01.

Copy/paste this into your R environment 

plotouup <- function(a){
# upwards simulation of ...
# Ornstein-Uhlenbeck process (simulated as a normal random walk of 1000 steps)
# the parameter  a  is the fraction of the way toward  0  that the
# value is moved at each step.
# If  a = 0   it is just a Brownian Motion,  a > 0  is OU
x <- rnorm(1000,0,sd=sqrt(0.001));
  y = rep(0,1000);
  y[1] = x[1];
  for (j in 2:1000) {
    y[j] = (1-a)*y[j-1]+x[j];
  }
yl <- 2.5;
t <- seq(0.001,1,0.001);
plot(y,t,type="l",xlim=c(-yl,yl),xlab="x",ylab="time");
for (i in 2:10) {
  x <- rnorm(1000,0,sd=sqrt(0.001));
  y[1] = x[1];
  for (j in 2:1000) {
    y[j] = (1-a)*y[j-1]+x[j];
    }
  lines(y,t,type="l");
  }
}