Brownian Motion Intuition

Brownian motion is what a random walk becomes as you shrink the step size toward zero while speeding up how often you step — the continuous- time limit of the discrete random walk from the last module. It's the foundation of the GBM stock-price model from the Monte Carlo course: GBM is literally "Brownian motion, exponentiated, with a drift term added."

import random, math

random.seed(0)
w = 0.0
dt = 0.01
for _ in range(100):
    z = random.gauss(0, 1)
    w += math.sqrt(dt) * z   # each increment ~ N(0, dt)
print(round(w, 4))

Each increment is itself normally distributed with variance dt — that sqrt(dt) scaling (not just dt) is the defining feature of Brownian motion, and it's exactly the same scaling used inside the GBM formula.

Here's the property that makes Brownian motion mathematically special: its variance at time T equals T itself, REGARDLESS of how many steps you used to simulate it. Take 10 big steps of size dt = T/10, or 10,000 tiny steps of size dt = T/10000 — the variance of the final position is the same either way, because each step's variance (dt) sums up to exactly T no matter how you slice it.

# 10 steps of size 0.1, OR 1000 steps of size 0.001 -- both simulate T=1
# and both have Var(W(1)) ≈ 1, not different values

This is why Brownian motion is the right continuous-time limit of a random walk — the discrete walk's variance already scaled linearly with step COUNT; Brownian motion makes that "linear in elapsed time" instead of "linear in step count," which is what lets you meaningfully talk about "the variance at time T" independent of simulation granularity.