Brownian motion and normal random increments

Brownian motion supplies the continuous random input used by an SDE. Its increments have a precise normal distribution even though individual paths look irregular.

Brownian motion is the source of continuous random fluctuation

Standard Brownian motion \(W(t)\) starts at \(W(0)=0\). Its path is continuous, but it changes irregularly and has no ordinary derivative.

\[\Delta W=W(t+\Delta t)-W(t).\]

\(\Delta W\) is the change in Brownian motion over an interval; it is not the value \(W(t)\) itself.

Distribution of an increment

\[\Delta W\sim N(0,\Delta t).\]

Property	Meaning
Mean 0	No systematic upward or downward direction.
Variance \(\Delta t\)	Longer intervals allow more variation.
SD \(\sqrt{\Delta t}\)	Typical increment size scales with the square root of time.
Independent disjoint increments	A new interval’s increment does not depend on earlier non-overlapping increments.

Why the path shown by a computer looks discrete

Brownian motion is continuous in the mathematical model. A computer stores it only at finitely many times and draws lines between those points. Reducing \(\Delta t\) reveals more detail, but it does not turn the model from discrete into continuous; it improves a discrete representation of a continuous path.

Accumulating increments

\[W(t_n)=\sum_{k=0}^{n-1}\Delta W_k.\]

Brownian motion is constructed by cumulatively adding independent increments. Increments fluctuate around zero, but their cumulative sum need not remain near zero on one path.

Interactive Python laboratory

The first graph shows six continuous-line representations of Brownian paths. The second checks the distribution of \(W(1)\) across 10,000 independent paths.

Interactive PythonBrownian paths and increments

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

T = 1.0
dt = 0.002
steps = int(T / dt)
seed = 15
rng = np.random.default_rng(seed)

times = np.linspace(0, T, steps + 1)

# Six displayed paths
dW = np.sqrt(dt) * rng.normal(
    0, 1, size=(6, steps)
)
W = np.column_stack([
    np.zeros(6),
    np.cumsum(dW, axis=1)
])

plt.figure(figsize=(9, 5))
for path in W:
    plt.plot(times, path, linewidth=1)

plt.axhline(0, color="black", linewidth=0.8)
plt.xlabel("Time")
plt.ylabel("W(t)")
plt.title("Six Brownian-motion paths")
plt.grid(alpha=0.25)
plt.tight_layout()

# Distribution at the fixed time T = 1
number_of_paths = 10000
increments = np.sqrt(dt) * rng.normal(
    0, 1, size=(number_of_paths, steps)
)
W_at_T = increments.sum(axis=1)

summary = pd.DataFrame({
    "quantity": ["Theoretical", "Simulated"],
    "mean of W(1)": [0, W_at_T.mean()],
    "variance of W(1)": [T, W_at_T.var(ddof=1)]
})

print(summary.round(4).to_string(index=False))

plt.figure(figsize=(9, 5))
plt.hist(W_at_T, bins=45, density=True,
         color="#79ae90", edgecolor="white")
x = np.linspace(-4, 4, 400)
normal_density = (
    np.exp(-(x ** 2) / 2) / np.sqrt(2 * np.pi)
)
plt.plot(x, normal_density, color="#b42318",
         linewidth=2, label="N(0, 1) density")
plt.xlabel("W(1)")
plt.ylabel("Density")
plt.title("Across paths, W(1) is normally distributed")
plt.legend()
plt.grid(alpha=0.25)
plt.tight_layout()

Output

Run the code to see the result.

Brownian motion is not biological population size

\(W(t)\) can be negative and has no direct interpretation as a number of infected people. In an SDE, the diffusion coefficient transforms its increment into a state-dependent perturbation of the biological variable.

What this lesson adds

You can now distinguish Brownian motion from its increments, state their distribution and independence, construct paths by cumulative addition, and explain why a computer plot is a grid-based representation of a continuous stochastic process.