Free lesson · Stochastic calc

Start with one random step before continuous noise

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Start with the idea

A random variable has several possible values.
Calculate its average and spread before introducing a random path.

Symbols, units & horizon

Z: dimensionless toy shock, equally likely −1 or +1
a: a fixed multiplier in the scaling rule
h: positive elapsed time in days
σ: shock scale in units per square root of day
ΔX: random change in units
E: probability-weighted mean
Var: mean squared deviation from the mean

When and why to use this

Use this scaling as preparation for simulation increments. The next lesson replaces the toy shock with the Gaussian increments of Brownian motion.

Toy shock Z: either −1 or +1, each with probability one half.
Mean $E [Z] = 0$ : positive and negative outcomes balance.
Variance $Var (Z) = 1$ : squared deviations are both 1.
Multiplying a shock by a scales its standard deviation by $| a |$ .
This two-outcome model teaches scaling. It is not a Gaussian Brownian increment.

Δ X = σ \sqrt{h} Z, E [Δ X] = 0, Var (Δ X) = σ^{2} h

Core rule · definition and worked arithmetic

Start with one random step before continuous noise

The two changes are $- σ \sqrt{h}$ and $+ σ \sqrt{h}$ . Equal probabilities give mean zero.
Square each deviation: both give $σ^{2} h$ . Their weighted average is therefore $σ^{2} h$ .
Standard deviation is the square root of variance: $σ \sqrt{h}$ for nonnegative σ.

Work it by hand

With σ=2 units/√day and h=.25 day, the change is −1 or +1 unit. Mean=0, variance=1 unit², standard deviation=1 unit.

Use the rule

Name the inputs and units.
Work the small example by hand.
Check the result before continuing to the next lesson.

Before moving on

Explain the core rule in one sentence, reproduce the worked calculation and solve both practice variations.

Research sources, review dates and limitations

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 math import sqrt

def toy_shock(scale, elapsed, sign):
    if scale < 0 or elapsed <= 0 or sign not in (-1,1):
        raise ValueError("Nonnegative scale, positive time and sign +/-1 required")
    return scale*sqrt(elapsed)*sign

print(toy_shock(2,.25,-1), toy_shock(2,.25,1))

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