Image:Girsanov.png
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| This is a file from the Wikimedia Commons. The description on its description page there is shown below. |
| Description |
Visualisation of the Girsanov theorem — The left side shows a Wiener process with negative drift under a canonical measure P; on the right side each path of the process is colored according to its likelihood under the martingale measure Q. The density transformation from P to Q is given by the Girsanov theorem. |
|---|---|
| Source |
created with GNU R, see source below |
| Date |
7. july 2006 |
| Author |
Martin Keller-Ressel (uploaded by Thomas Steiner) |
| Permission |
Martin Keller-Ressel put it under the GFDL |
R-source code:
set.seed(145)
par("mar"=c(4,2,2,2))
npath <- 30
N <- 1000
t <- seq(0,1,length=N+1)
sigma = 0.5
nu = - 0.7
hue = 0.03
dampen = 0.25
BM.norm <- matrix(rnorm(npath*N,sd=1/sqrt(N),mean=0),ncol = npath)
BM.norm <- rbind(0,apply(BM.norm,2,cumsum))
tmat <- matrix(rep(t,npath),ncol=npath,byrow=FALSE)
BM <- sigma*BM.norm + tmat*nu
weights.orig <- matrix(1,ncol=npath,nrow=1)
png(file="girsanov.png",height=714,width=1212)
par(mfrow=c(1,2))
plot.new()
plot.window(xlim = c(0,1),ylim=range(BM))
col.orig <- hsv(h=hue,s=0.6*weights.orig,v=.95)
for(i in 1:npath) {
lines(t,BM[,i],cex=1.5,col=col.orig[i])
}
box(col="grey",cex=1.5)
title(sub="30 paths of a Brownian motion with negative drift",line=2,col.sub="grey10")
girsanov.weights <- exp(dampen * (-nu/sigma*BM.norm[N,] - 0.5*(nu/sigma)^2))
girsanov.weights <- (girsanov.weights - min(girsanov.weights)) / diff(range(girsanov.weights))
col.girsanov <- hsv(h=hue,s=girsanov.weights,v=0.95)
plot.new()
plot.window(xlim = c(0,1),ylim=range(BM))
for(i in 1:npath) {
lines(t,BM[,i],cex=1.5,col=col.girsanov[i])
}
title(sub="The same paths reweighted according to the Girsanov formula",line=2,col.sub="grey10")
box(col="grey",cex=1.5)
dev.off()
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