A (standard) Wiener process is a continuous-time stochastic process \(\{W(t)\}_{t\ge0}\) with initial value \(W(0)=0\) and instantaneous increments
$$\newcommand{\der}{\mathrm{d}} \der W(t)\sim N(0,\der t).$$
We can simulate such a process as follows.
First, create a sequence of times \(t\) at which to store the value of \(W(t)\):
t_max = 100
dt = 1e-2
t = seq(0, t_max, by = dt)
Increasing t_max creates a longer path, while decreasing dt creates a smoother path.
Now simulate the random increments and take their cumulative sum:
dW = rnorm(length(t) - 1, mean = 0, sd = sqrt(dt))
W = c(0, cumsum(dW))
Here are three sample paths generated by this procedure:
We can use \(\{W(t)\}_{t\ge0}\) to construct an Ornstein-Uhlenbeck process \(\{X(t)\}_{t\ge0}\).
This process has instantaneous increments
$$\der X(t)=-\theta X(t)\der t+\der W(t),$$
where \(\theta\ge0\) controls the process’ tendency to mean-revert.
We can compute its values \(X(t)\) by iterating over dW:
theta = 1
X = rep(0, length(dW))
i = 1
while (i < length(dW)) {
X[i + 1] = X[i] - theta * X[i] * dt + dW[i + 1]
i = i + 1
}
The chart below compares the sample paths obtained using different \(\theta\) values.
Each path uses the same realization of the underlying Wiener process \(\{W(t)\}_{t\ge0}\).
If \(\theta=0\) then \(X(t)=W(t)\) for all \(t\ge0\).
The mean magnitude of \(X(t)\) falls as \(\theta\) rises because this makes the process more mean-reverting.
