Free lesson · Stochastic calc

Before Itô: random walks, Brownian motion and information

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Start with the idea

Stochastic calculus studies functions of random processes.
Begin with a random walk whose increments have zero mean and variance proportional to elapsed time.
Brownian motion is an idealised continuous-time limit with continuous but extremely irregular paths.

Symbols, units & horizon

W_t: standard Brownian value at time t
h: time-step duration
ΔW_k: Brownian increment over step k
Z_k: independent standard normal draw
N(0,1): normal distribution with zero mean and unit variance
n: number of steps
T=nh: elapsed time
Var: variance
SD: standard deviation, the square root of variance
W has square-root-time units and Z is dimensionless
𝓕_t: information available through t
∼: distributed according to

When and why to use this

Use increment scaling and information timing before interpreting any diffusion simulation. This prepares the GBM, Itô and stochastic-volatility lessons that follow.

Suggested preparation: ordinary derivatives, integrals, partial derivatives, and probability.

Standard Brownian motion starts at zero, has independent Gaussian increments, and an increment over length h has mean zero and variance h.
Its typical scale is therefore √h, not h.
The paths are almost surely nowhere differentiable; ordinary velocity intuition fails.
A filtration is the information available through time.
An adapted process uses current and past information, and an Itô trading integrand must satisfy the relevant non-anticipation and integrability requirements.
This is a mathematical version of not trading on future shocks.

Δ W_{k} = \sqrt{h} Z_{k}, Z_{k} \sim 𝒩 (0,1), W_{T} = \sum_{k = 1}^{n} Δ W_{k}, Var (W_{T}) = n h = T

Calculus: derivation and arithmetic

Before Itô: random walks, Brownian motion and information

Multiplying a unit-variance Z by √h gives variance h, since variance scales by the square of the multiplier.
Independent increments have additive variances. Summing n increments therefore gives variance nh=T and mean zero.
If you used hZ instead, the total variance would be nh²=Th, which tends to zero as the grid is refined. That scaling would remove the intended randomness.

Work it by hand

With T=1 and n=100, each increment has SD .1 and variance .01. The sum has variance 1 and SD 1, not SD 10.

An analogy to remember

Many tiny, independent pushes can accumulate into a persistent random displacement even as each individual push gets smaller. Their variances add; their signed values do not all push in the same direction.

How this becomes a building block

Brownian increments are a noise component inside a diffusion model. Adding a drift and state-dependent scale produces an SDE. A random walk is not itself a complete market model; jumps, microstructure and changing volatility may require other assumptions.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

from random import Random
from math import sqrt

def brownian_path(horizon,steps,seed=7):
    if horizon<0 or not isinstance(steps,int) or steps<1: raise ValueError("Valid horizon and positive steps required")
    rng=Random(seed)
    h=horizon/steps
    path=[0.0]
    for _ in range(steps): path.append(path[-1]+sqrt(h)*rng.gauss(0,1))
    return path

print(brownian_path(1,4))

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