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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,lwd=2,ylab="",xlab="Zeit",main="Hitting time von drei
Sampels der Brownschen Bewegung als Beispiel für eine Stoppzeit") lines(t,bm2,col=cs,lwd=2) lines(t,bm3,col=cs,lwd=2) abline(h=a,col=cs,lty=1,lwd=3) text(x=tau1*T/N,y=a*1.14, labels = substitute(tau==t1,list(t1=tau1*T/N)),pos=4,col=cs,cex=1.5) text(x=tau2*T/N,y=a*1.14, labels = substitute(tau==t2,list(t2=tau2*T/N)),pos=4,col=cs,cex=1.5) text(x=T,y=a*1.14, labels = substitute(tau>T,list(T=T)),pos=2,col=cs,cex=1.5) dev.off()
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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) bm1=a bm2=c(0,cumsum(rnorm(N,mean=0,sd=sqrt(T/N)))) tau2=which(bm2>a) bm2=a bm3=c(0,cumsum(rnorm(N,mean=0,sd=sqrt(T/N)))) tau3=which(bm3>a) bm3=a
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Hitting time von drei
Sampels der Brownschen Bewegung als Beispiel für eine Stoppzeit.
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Hitting times and stopping times of three samples of brownian motion
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