Wiener Process: Interactive Simulators, Scale Tests, and Statistical Diagnostics

This page provides a compact and rigorous presentation of the Wiener process (Brownian motion): mathematical definition and main properties, practical guidance for numerical simulation, and tools for statistical verification. Interactive simulations are available to generate paths, compute the mean-squared displacement (MSD) and analyze the distribution of increments, accompanied by automated diagnostic tests (MSD log-log fit, KS test, QQ preparation). A simple agent-based model (ABM) is also included to compare interaction effects with Gaussian diffusion; the material is intended for teaching and engineering applications, with guidelines for reproducibility and numerical validation.

Definition

\[ W_0 = 0,\qquad W_t - W_s \sim \mathcal{N}(0,\,t-s),\quad\text{(independent increments)}. \]

Practical meaning:

  • Gaussian increments: the amount the process changes over an interval of length \( \tau \) is \( \sim \mathcal{N}(0,\tau) \).
  • Independence: increments on disjoint intervals are independent — useful to sum variances.
  • Time scaling: \(\mathrm{Var}(W_t)=t\) → standard deviation ≈ \(\sqrt{t}\). In practice uncertainty grows with the square root of time.

To simulate: pick a step \(\Delta t\), generate \(\Delta W_i = \sqrt{\Delta t}\,Z_i\) with \(Z_i\sim N(0,1)\) independent, and accumulate.

Properties of the Wiener process

The Wiener process has structural properties that determine its behavior both theoretically (Gaussian process, martingale, random-walk limit) and in applications, where it serves as the basic diffusion model and as the stochastic term in SDEs. The items below — linear variance, irregularity, quadratic variation, approximation by random walks and role in stochastic equations — form the core of results explaining why Wiener is the fundamental "noise" in continuous modelling.

Linear variance in time

For a Wiener process the increments on disjoint intervals are independent and have variance equal to the interval length. Splitting \([0,t]\) into \(n\) intervals of width \(\Delta t = t/n\), each increment has variance \(\Delta t\). Summing variances by independence yields \(\mathrm{Var}(W_t)=t\). This is the diffusion property with scaling \(\sqrt{t}\).

Irregularity and non-differentiability

Trajectories of the Wiener process are continuous but almost surely non-differentiable. Intuitively, the ratio \(\dfrac{W_{t+\Delta t}-W_t}{\Delta t}\) has variance of order \(1/\Delta t\), which diverges as \(\Delta t\to 0\). The local "acceleration" is thus too irregular to admit a classical derivative.

Unit quadratic variation

\[ \sum_{i}(W_{t_{i+1}} - W_{t_i})^2 \xrightarrow{|\Pi|\to 0} T. \]

The Wiener process has quadratic variation exactly equal to the interval length. This property is characteristic: it distinguishes Wiener from regular functions and underlies Itô calculus, where terms \((\Delta W)^2\) produce contributions of order \(\Delta t\).

Approximation by random walk (Donsker's theorem)

A Wiener process can be obtained as the scaled limit of a random walk. Donsker's theorem extends the CLT to entire processes: the interpolated path of a random walk, rescaled in time and amplitude, converges in distribution to the Wiener process. This justifies simulations based on sums of Gaussian or binary increments.

Connection with SDEs and Euler–Maruyama

Wiener is the fundamental "noise" in stochastic differential equations \[ dX_t = a(X_t,t)\,dt + b(X_t,t)\,dW_t. \] The Euler–Maruyama scheme \[ X_{n+1} = X_n + a(X_n,t_n)\,\Delta t + b(X_n,t_n)\,\Delta W_n \] uses Wiener increments directly, \( \Delta W_n \sim \mathcal{N}(0,\Delta t) \). The structural properties of Wiener also determine convergence orders: strong \(O(\Delta t^{1/2})\), weak \(O(\Delta t)\).

Interactive simulator

Diagnostic report: run Simulate then Run diagnostics.
Paths: \(X_t=\mu t + \sigma W_t\); dashed line = theoretical mean \(\mu t\).
Empirical MSD (average over paths) with theoretical reference \(\sigma^2 t\).
Histogram of increments \(\Delta X\) and scaled theoretical normal density.

Charts explanation

Objectives

  • Check that paths are centered on \(\mu t\) (correct mean).
  • Verify time scaling: the MSD should grow like \(\sigma^2 t\) (log-log fit with slope ≈ 1).
  • Check that increments \(\Delta X\) are Gaussian with mean \(\mu\Delta t\) and variance \(\sigma^2\Delta t\) (histogram, KS, QQ).

1) Initial settings (example)

For a quick check: set μ = 0, σ = 1, T = 1, n = 1000, m = 200. Then click Simulate paths & MSD.

2) Interpretation of the three charts

  • Paths: the cloud of paths should be symmetric around the dashed line \(\mu t\). Systematic offsets indicate incorrect drift or RNG/seed issues.
  • MSD (mean squared displacement): the empirical curve (mean over paths) should follow the reference line \(\sigma^2 t\). In log-log scale the expected slope is ≈ 1.
  • Increments histogram: the histogram shape should overlap the scaled theoretical normal density. Localized differences or heavy tails suggest departures from Gaussianity.

3) Automated numerical diagnostics (the Run diagnostics button)

After simulation, the automatic report runs the following tests and produces readable indicators:

  • MSD log-log fit: estimates the slope and the coefficient of determination R². Quick interpretation:
    • |slope−1| ≤ 0.05 → slope consistent with normal diffusion.
    • 0.05 < |slope−1| ≤ 0.12 → moderate deviation; consider increasing m or reducing Δt.
    • |slope−1| > 0.12 → anomaly (possible ABM interactions or numerical error).
  • Mean/variance comparison of increments: numeric comparison between empirical mean and \(\mu\Delta t\), empirical variance and \(\sigma^2\Delta t\). Small differences are expected for finite samples.
  • KS test: computes the statistic D comparing the increments' distribution to the theoretical normal; the report shows D and the critical threshold (α = 0.05). If D ≤ crit → do not reject Gaussianity.
  • QQ data: a set of theoretical vs empirical points is prepared and logged to the console for visual inspection (useful to observe tails or systematic deviations).

4) What to do if results are unexpected

  • If the mean of the paths is shifted: check the value of μ and the RNG seed; repeat the simulation with a larger m.
  • If the MSD slope is too low/high: try increasing the number of paths m (reduces MSD noise) or reducing the time step Δt (reduces numerical error). For the ABM expect deviations due to interactions.
  • If the histogram shows heavy tails: check for outliers, verify that σ and Δt are correct; use more samples for the KS test.

5) Best practices

  • Expected MSD slope ≈ 1 in log-log; practical tolerance ±0.05 for moderate samples.
  • KS: approximate 5% critical threshold is \( \dfrac{1.36}{\sqrt{N}} \) (where N is the number of sampled increments).
  • For reliable tests gather at least a few hundreds of increments (preferably 1000+ when possible).

References

  1. Karatzas & Shreve — Brownian Motion and Stochastic Calculus.
  2. Kloeden & Platen — Numerical Solution of Stochastic Differential Equations.
  3. Billingsley — Convergence of Probability Measures (Donsker theorem).