Simulating Wiener and Ornstein-Uhlenbeck processes

A (standard) Wiener process is a continuous-time stochastic process $\left\{W\right(t){\}}_{t\ge 0}$ with initial value $W\left(0\right)=0$ and instantaneous increments $$dW\left(t\right)\sim N(0,dt).$$ We can simulate such a process as follows. First, create a sequence of times $t$ at which to store the value of $W\left(t\right)$:

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 $\left\{W\right(t){\}}_{t\ge 0}$ to construct an Ornstein-Uhlenbeck process $\left\{X\right(t){\}}_{t\ge 0}$. This process has instantaneous increments $$dX\left(t\right)=-\theta X\left(t\right)dt+dW\left(t\right),$$ where $\theta \ge 0$ controls the process’ tendency to mean-revert. We can compute its values $X\left(t\right)$ 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 $\left\{W\right(t){\}}_{t\ge 0}$. If $\theta =0$ then $X\left(t\right)=W\left(t\right)$ for all $t\ge 0$. The mean magnitude of $X\left(t\right)$ falls as $\theta$ rises because this makes the process more mean-reverting.