This semester my studies all involve one key mathematical object: Gaussian processes. I’m taking a course on stochastic processes (which will talk about Wiener processes, a type of Gaussian process and arguably the most common) and mathematical finance, which involves stochastic differential equations (SDEs) used for derivative pricing, including in the Black-Scholes-Merton equation. Then I’m involved in a Gaussian process and stochastic calculus reading group. So these processes will take up a lot of my attention.

In a conversation about these processes with a fellow graduate student I was explaining the idea that different kernels (covariance functions, or ) define different Gaussian processes and simply changing the kernel will produce new processes with completely different properties. Let be the kernel of a process. is the kernel associated with the Wiener process and produces a process that is continuous everywhere but not differentiable anywhere, and with independent, Gaussian-distributed increments. On the other hand, the process defined by the kernel is not only continuous but differentiable everywhere, yet does not have independent increments.

I wanted to drive home the point that different kernels yield processes with wildly different properties by simulating and plotting them on a computer. So I whipped out the following R function in less than ten minutes (not counting documentation), and it does exactly what I want it to do.

library(MASS)

gaussprocess <- function(from = 0, to = 1, K = function(s, t) {min(s, t)},
                         start = 0, m = 1000) {
  # Simulates a Gaussian process with a given kernel
  #
  # args:
  #   from: numeric for the starting location of the sequence
  #   to: numeric for the ending location of the sequence
  #   K: a function that corresponds to the kernel (covariance function) of
  #      the process; must give numeric outputs, and if this won't produce a
  #      positive semi-definite matrix, it could fail; default is a Wiener
  #      process
  #   start: numeric for the starting position of the process
  #   m: positive integer for the number of points in the process to simulate
  #
  # return:
  #   A data.frame with variables "t" for the time index and "xt" for the value
  #   of the process
  
  t <- seq(from = from, to = to, length.out = m)
  Sigma <- sapply(t, function(s1) {
    sapply(t, function(s2) {
      K(s1, s2)
    })
  })
  
  path <- mvrnorm(mu = rep(0, times = m), Sigma = Sigma)
  path <- path - path[1] + start  # Must always start at "start"
  
  return(data.frame("t" = t, "xt" = path))
}

Below are example processes simulated by this function.

(Wiener process)

(Gaussian kernel)

(Something completely different)

Hopefully you found this code snippet entertaining, if not useful.

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