Derivatives Portfolio & Risk Fixed Income Corporate Finance Probability & Measure Stochastic Models Market Microstructure Guide

Stochastic Models

Random Walk lab Symmetric Random Walk Rescaled Random Walk Standard deviation Central Limit Theorem

Brownian Motion

Brownian lab Defining the Wiener process Increment explorer Joint distributions and covariance Brownian transformations and invariance Quadratic variation Brownian bridge Running maximum and minimum Reflection principle First passage times Hitting probabilities

Geometric Brownian Motion

Lab Definition: ABM vs GBM Euler–Maruyama Mean reversion Jumps Correlated motions

Lab Itô integral Itô's lemma Martingales Girsanov

Lab Variance reduction Feynman–Kac

Connected Guides

Random walks and Brownian motion
The law of large numbers and the central limit theorem
Probability spaces, σ-algebras and random variables

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Rescaled Random Walk

What happens when we compress the walk in time and scale it by √n?

Goal: watch the paths refine and the terminal distribution approach the standard normal as n grows.

Simulation controls

Refinement level nStart and end stay fixed; each notch doubles the steps, inserting new points between the existing ones

Number of steps1

1162564,096

Overlaid paths4

16

Random seed7

19999

W⁽ⁿ⁾(1) statistics

Empirical mean

0.040

Theoretical mean

0.000

Empirical variance

0.999

Theoretical variance

1.000

The terminal variance is ≈ 1 for every n and the mean ≈ 0, with the distribution approaching the standard normal (Donsker). Individual paths, however, do not converge to one function — they get rougher.

Rescaled paths W⁽ⁿ⁾(t)

n = 1

Typical step size 1/√n ≈ 1.000; the endpoints never move, refinement only adds detail between them.

Terminal distribution vs N(0,1)

Empirical densityN(0,1) density