Image:HittingTimes1.png
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| This is a file from the Wikimedia Commons. The description on its description page there is shown below. |
| Description |
some stopping times (even hitting times!) of brownian motion |
|---|---|
| Source |
with the help of GNU R statistics / math software. see the source below |
| Date |
8 Apr. 2006 |
| Author | |
| Permission |
Thomas Steiner put it under the GFDL |
R-source code:
T=7
N=100000
set.seed(303898)
a=0.9
cs=c("orange", "navy", "seagreen4", "grey31")
bm1=c(0,cumsum(rnorm(N,mean=0,sd=sqrt(T/N))))
tau1=which(bm1>a)[1]
bm1[tau1:(N+1)]=a
bm2=c(0,cumsum(rnorm(N,mean=0,sd=sqrt(T/N))))
tau2=which(bm2>a)[1]
bm2[tau2:(N+1)]=a
bm3=c(0,cumsum(rnorm(N,mean=0,sd=sqrt(T/N))))
tau3=which(bm3>a)[1]
bm3[tau3:(N+1)]=a
png(filename = "HittingTimes1.png", width=1200, height=900, pointsize = 12)
par(bg="gray88")
t=seq(0,T,length=N+1)
plot(t,bm1,type="l",ylim=range(bm1,bm2,bm3,a*1.2),col=cs[1],lwd=2,ylab="",xlab="Zeit",main="Hitting time von drei Sampels der Brownschen Bewegung als Beispiel für eine Stoppzeit")
lines(t,bm2,col=cs[2],lwd=2)
lines(t,bm3,col=cs[3],lwd=2)
abline(h=a,col=cs[4],lty=1,lwd=3)
text(x=tau1*T/N,y=a*1.14, labels = substitute(tau[2]==t1,list(t1=tau1*T/N)),pos=4,col=cs[1],cex=1.5)
text(x=tau2*T/N,y=a*1.14, labels = substitute(tau[1]==t2,list(t2=tau2*T/N)),pos=4,col=cs[2],cex=1.5)
text(x=T,y=a*1.14, labels = substitute(tau[3]>T,list(T=T)),pos=2,col=cs[3],cex=1.5)
dev.off()
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