📈 Wiener Process Dynamics & Stochastic Drift

About this simulation

This simulator plots sample paths of a Wiener process, W(t), the continuous-time random walk whose increments are independent and normally distributed with variance equal to the elapsed time. It also lets you switch to two derived models: the mean-reverting Ornstein-Uhlenbeck process and Geometric Brownian Motion, the stochastic differential equation behind the Black-Scholes option-pricing formula. Paths are generated numerically with the Euler-Maruyama scheme, the standard way to integrate a stochastic differential equation on a computer.

Frequently asked questions

Frequently Asked Questions

• What is a Wiener process?
  A Wiener process (also called standard Brownian motion) is a continuous-time stochastic process W(t) with independent, normally distributed increments: W(t)−W(s) ~ N(0, t−s). It is the mathematical formalisation of the random jiggling observed by botanist Robert Brown in 1827 and later given rigorous form by Norbert Wiener in the 1920s.
• What is the Euler-Maruyama method?
  The Euler-Maruyama method is the stochastic analogue of Euler's method for ODEs. For an SDE dX = f(X,t)dt + g(X,t)dW, it approximates X(t+dt) ≈ X(t) + f(X,t)⋅dt + g(X,t)⋅√dt⋅N(0,1). The simulator uses this discretisation with a small time step dt to generate realistic sample paths.
• How does Geometric Brownian Motion relate to the Black-Scholes model?
  GBM is the stochastic differential equation dS = μS⋅dt + σS⋅dW, where μ is the drift (expected return) and σ is the volatility. The Black-Scholes option-pricing formula assumes that the underlying asset price follows GBM. Because GBM always stays positive and produces log-normally distributed prices, it is the canonical model for equity prices.
• What makes the Ornstein-Uhlenbeck process different from standard Brownian motion?
  The OU process adds a mean-reversion term: dX = θ(μ−X)dt + σdW. The parameter θ > 0 pulls the process back toward the long-run mean μ. Unlike pure Brownian motion, the OU process is stationary and its variance does not grow without bound. It is used to model interest rates (Vasicek model), commodity prices, and neuronal membrane potentials.
• What is the quadratic variation of a Wiener process?
  The quadratic variation of W over [0,T] equals T almost surely. This non-zero quadratic variation is what distinguishes stochastic calculus from ordinary calculus and necessitates Itô's lemma, which adds a correction term (½σ²f''dt) absent from the classical chain rule.
• Why do paths of the Wiener process look jagged?
  Wiener process paths are almost surely nowhere differentiable. At every point the function changes direction infinitely often — the local variation is infinite. This mathematical roughness, visible as the jagged zig-zag in the simulator, is captured quantitatively by the non-zero quadratic variation.
• What is the Itô integral and why does it differ from Riemann integration?
  The Itô integral ∫₀ᵀf(t)dW(t) is defined as a limit of sums that always use the left endpoint of each interval (non-anticipating). Because dW has finite quadratic variation, the ordinary Riemann-Stieltjes integral is not well-defined, and the Itô correction term ½σ²f'' appears when applying the chain rule (Itô's lemma).
• How many paths should I simulate to get reliable statistics?
  For rough statistics (mean path, approximate variance band) 20–50 paths are usually sufficient. For precise tail probabilities or option price estimates, Monte Carlo convergence scales as 1/√N, so 1 000–10 000 paths are typical in practice. The simulator draws up to 30 independent paths simultaneously so you can see the spread visually.
• What is the mean squared displacement of Brownian motion?
  For a standard Wiener process, E[W(t)²] = t. More generally, for d-dimensional Brownian motion with diffusion coefficient D, the mean squared displacement E[|r(t)|²] = 2dDt grows linearly with time — a key signature that distinguishes normal diffusion from anomalous (sub- or super-) diffusion.
• Can I export the simulated paths?
  The simulator runs entirely in the browser and does not send data to a server. You can take a screenshot of the canvas. For large-scale Monte Carlo work, running the Euler-Maruyama algorithm in Python with NumPy or in Julia with DifferentialEquations.jl is recommended.