File:Gamma-OU.png

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R-source Code:

######################################################
### OU-levy processes ################################
# trajectory of an OU-process, driven by some levy process (bdlp; here cauchy or gamma)
## starting at time t=0 at Z_0=Z0 until time t=T in "steps" equidistant steps; mean reversion "mr"
## parameters of the distribution of the bdlp are stored in param
process.ou<-function(param=c(0,1),mr=1,T=1,steps=500,bdlp="cauchy",Z0=0) {
  dt=T/(steps-1)
  ou=array(Z0,dim=steps)
  if (bdlp=="cauchy") {
    dLt=rcauchy(steps-1, location=param[1], scale=param[2]*dt)
  } else if (bdlp=="gamma") {
    dLt=rgamma(n=steps-1, shape=param[1], scale=param[2]*dt)
  }
  for (tm in 1:(steps-1)) {
    ou[tm+1]=ou[tm]-mr*ou[tm]*dt+dLt[tm]
  }
  return (list(time=seq(0,T,length=steps),Lt=c(0,cumsum(dLt)),ou=ou))
}

set.seed(29011980)
time=seq(0,5.7,length=5000)

png("Gamma-OU.png",width=1200, height=900)
par(mfrow=c(2,1))

gou=process.ou(param=c(2.1,0.9),mr=1.3,T=max(time),steps=length(time),bdlp="gamma",Z0=1.1)
plot(gou$time,gou$Lt,type="p",pch=".",main="Gamma process",col="purple",xlab="",ylab="")
plot(gou$time,gou$ou,type="p",pch=".",main="Gamma-OU process",col="purple",xlab="",ylab="")
abline(h=2.1*0.9/1.3,lty="dotted",col="darkgrey") #mean of limit-distribution
dev.off()

 

This chart was created with R.