File:Girsanov.png

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()