Image:OrnsteinUhlenbeck3.png

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R-Quelltext:

#Orenstein-Uhlenbeck Prozess mit normalverteiltem Anfangswert
ou<-function(theta=1, mu=1.2, sigma=0.3, init=c(mu, sigma^2/(2*theta)), from=0, to=2, steps=500*(to-from)) {
  t<-seq(from,to,length=steps)
  oup<-rnorm(n=1,mean=init[1],sd=sqrt(init[2]))
  for (i in (1:(steps-1))) {
    oup[i+1]=oup[i] + theta*(mu-oup[i])*(to-from)/steps+sigma*rnorm(n=1,mean=0,sd=sqrt(sigma^2*(to-from)/steps))
  }
  return(oup)
}

cols=c("navy", "darkolivegreen4", "red")
t<-seq(0,2,length=1000)
set.seed(311051)

png(filename="OrnsteinUhlenbeck3.png", width=1200, height=900, pointsize=12)
par(bg="lightgrey")
matplot(t,ou(init=c(0,0)),type="l", xlab="", ylab="",col=cols[1], lwd=2, ylim=c(0,2))
lines(t,ou(init=c(2,0)),col=cols[2], lwd=2)
lines(t,ou(),col=cols[3], lwd=2)
title(main="Ornstein - Uhlenbeck", cex.main=2.5)
text(x=1.85,y=0,expression(list(theta==1, mu==1.2, sigma == 0.3)) )
text(x=1.55,y=1.75,expression(paste("d",X[t]==theta(mu-X[t]),"dt+",sigma,"d", W[t])),cex=2.5)
text(x=0,y=0.07, expression(X[0]==0),col=cols[1])
text(x=0,y=1.92, expression(X[0]==2),col=cols[2])
text(x=0.03,y=1.29, expression(paste(X[0], "~N(", mu,",", frac(sigma^2,2*theta), ")") ),col=cols[3])
abline(h=1.2, lty=3, col="grey25")
dev.off()

• Ornstein-Uhlenbeck process