File:Comp-Comp-Poisson-Pro.png

de: Pfade von zwei kompensierten zusammengesetzten Poisson-Prozessen. Wie in de:Poissonprozess.png ist die Intensität (Sprunghäufigkeit) des blauen Prozesses mit 2,4 genau viermal so hoch wie die des roten Prozesses. Im gezeichneten Intervall [0,35] springt der blaue Prozess 66-mal, der rote 16-mal, also etwa viermal so oft. Bei beiden Prozessen sind die Sprünge normalverteilt mit Mittel 0,25. Diese Sprünge nach oben werden durch den negativen Drift genau so ausgeglichen (kompensiert), dass beide Prozesse Martingale sind. Da der blaue Prozess öfter nach oben springt, ist sein negativer "Drift" stärker.

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R source code:

set.seed(29011980)

##plots a compound (normal), compensated poisson process.
poisson.process<-function(lambda, T, mu, sigma, drift=-lambda*mu) {
  N=rpois(n=1, lambda=lambda*T)                                   #number of jumps
  t=sort(array(data=runif(n=N, min=0,max=T),dim=N))               #jump times
  y=array(data=rnorm(n=N,mean=mu,sd=sigma), dim=N)                #jump sizes
  t=t-t[1]
  s=array(NA, dim=2*N)
  s[2*(1:N)-1]=diff(c(t,T))*drift
  s[2*(1:N)]  =y
  s=c(0,cumsum(s))
  ti=seq(from=1, to=N)
  return(list(s=s,t=t,ti=ti))
}

pp1=poisson.process(lambda=2.4,T=35,mu=0.25,sigma=1)
pp2=poisson.process(lambda=0.6,T=35,mu=0.25,sigma=1)

png(filename="Comp-Comp-Poisson-Pro.png", width=1600, height=1200, pointsize=12)
par(bg="whitesmoke")
plot(c(0,pp1$t),c(0,pp1$t),type="n",xlab="t",ylab="",ylim=range(pp1$s,pp2$s))
segments(pp1$t[pp1$ti], pp1$s[2*pp1$ti-1], pp1$t[pp1$ti+1], pp1$s[2*pp1$ti], col="blue", lwd=2)
segments(pp2$t[pp2$ti], pp2$s[2*pp2$ti-1], pp2$t[pp2$ti+1], pp2$s[2*pp2$ti], col="red", lwd=2)

title(main="Pfade von kompensierten zusammengesetzten Poisson-Prozessen",cex.main=1.7)
legend(x="bottomleft", inset=0.01, cex=1.4, title="Intensität", legend=c(expression(lambda[1]==2.4,lambda[2]==0.6)), col=c("blue","red"), lwd=3) 
text(x=par("usr")[1]*0.9, y=par("usr")[4]*0.95, labels="Sprunghöhe ~ N(0.25,1)", adj=c(0,1))
dev.off()

 

This chart was created with R.