#### Brownian Motion - Normal increments

t<-seq(0,1,0.001); 
x<-matrix(0,nrow=1001,ncol=3);
x[,1]<-t; 
W<-rnorm(1000,0,1)
x[2:1001,2]<-W
for (i in 2:1001) { x[i,3]<-x[i-1,3]+x[i,2]}
plot(x[,1],x[,3],type="l",lwd=2)


#### Brownian Motion - uniform increments

t<-seq(0,1,0.001); 
x<-matrix(0,nrow=1001,ncol=3);
x[,1]<-t; 
W<-runif(1000,-5,5)
x[2:1001,2]<-W*sqrt(3/25)
for (i in 2:1001) { x[i,3]<-x[i-1,3]+x[i,2]}
plot(x[,1],x[,3],type="l",lwd=2)


############# Brownian Motion Many Paths #########################

t<-seq(0,1,0.001); number_of_paths=5
x<-matrix(0,nrow=1001,ncol=number_of_paths+1);
x[,1]<-t; 

for (i in 1:number_of_paths){
     W<-rnorm(1000,0,1)
     for (j in 2:1001){
          x[j,i+1]<-x[j-1,i+1]+W[j-1]
         }
     }
plot(x[,1],x[,2],type="l",lwd=2,col="1",ylim=c(-80,80))
for (i in 1:number_of_paths-1){ lines(x[,1],x[,i+2],type="l",lwd=2,col=i+1)}

#### Brownian Bridge - Normal increments

t<-seq(0,1,0.001); 
x<-matrix(0,nrow=1001,ncol=3);
x[,1]<-t; 
W<-rnorm(1000,0,1)
x[2:1001,2]<-W
for (i in 2:1001) { x[i,3]<-x[i-1,3]+x[i,2]}
plot(x[,1],x[,3]-x[,1]*x[1001,3],type="l",lwd=2)
abline(0,0, col="red",lwd=2)


###################### Paley - Wiener Series ##############################

t<-seq(0,1,0.001); N<-1000
x<-matrix(0,nrow=1001,ncol=3);
x[,1]<-t; 
x[,2]<-sqrt(2*pi)*t*rnorm(1,0,1)
plot (x[,1],x[,2],type="l",lwd=1,ylim=c(-5,5))
for (i in 1:N){ y<-2*rnorm(1,0,1)*(sin(pi*i*t))/(i*sqrt(pi))
                x[,2]<-x[,2]+y
                #lines(x[,1],x[,2],type="l",lwd=1)
                }
lines(x[,1],x[,2],type="l",lwd=1)