Wiener Process Simulation

This page presents a simple interactive simulation of a Wiener process, also known as Brownian motion. The Wiener process is a fundamental stochastic model used in mathematics, physics, and finance to describe random continuous-time motion. By generating small random increments and accumulating them over time, the simulation produces a sample path that illustrates how unpredictable and irregular such motion can be. This visualization helps demonstrate key properties of Brownian motion, including continuity, randomness, and the way variance grows with time.

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Wiener Process Definition

A Wiener process W(t) satisfies the SDE:

$$dW(t) = \sigma \, dB(t), \quad W(0) = 0$$

• B(t) is standard Brownian motion.
• \(\sigma\) is the volatility (for a standard Wiener process, \(\sigma = 1\)).

In discrete time (Euler–Maruyama method):

$$W_{t+\Delta t} = W_t + \sigma \sqrt{\Delta t} \, Z_t​$$

• \(Z_t \sim \mathcal{N}(0,1)\) are independent standard normal random variables.