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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 $E[X_t X_s]$) define different Gaussian processes and simply changing the kernel will produce new processes with completely different properties. Let $K(s, t)$ be the kernel of a process. $K(s, t) = \min(s, t)$ 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 $K(s, t) = e^{-(s - t)^2}$ 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 + start  # Must always start at "start"

return(data.frame("t" = t, "xt" = path))
}


Below are example processes simulated by this function. $K(s, t) = \min(s, t)$ (Wiener process)  $K(s, t) = e^{-16(s - t)^2}$ (Gaussian kernel)  $K(s, t) = \frac{1}{1 + \left|s - t \right|}$ (Something completely different) Hopefully you found this code snippet entertaining, if not useful.

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