What is a possible R code to simulate a one-dimensional Brownian Motion path with each step explained?

Question

n <- 1000
t <- 100
bm <- c(0, cumsum(rnorm(n,0,sqrt(t/n))))
steps <- seq(0,t,length=n+1)
plot(steps,bm,type="l",xlab="Time",ylab="B(t)")

This is what I have managed to do but the steps do not make total sense.

• Why must we define n and t?
• Why do we separate the 0 from the cumsum?
• Why we use the function cumsum?
• What is the significance of (n,0,sqrt(t/n))?
• Why do we define steps?

How would this code be modified to simulate a two-dimensional Brownian motion path or several Brownian motions?

Answer 1

• Why must we define n and t?

Because they are parameters of your problem, and if you need to change it you just change the value in one point of your code

• Why do we separate the 0 from the cumsum?

Because is your starting point. The starting point is 0, not a random one.

• Why we use the function cumsum?

Because in order to know where the process is at the time t, you need to sum all the preceeding steps.

• What is the significance of (n,0,sqrt(t/n))?

rnorm(n, 0, sqrt(t/n)) indicates a random vector with normal distribution of mean n and std dev of sqrt(t/n).

• Why do we define steps?

To have a process that is long as we prefer