SDE STOCHASTIC PATHS

Brownian Motion, GBM & Ornstein-Uhlenbeck

Stochastic differential equations model systems driven by random noise. Multiple sample paths are drawn simultaneously. The shaded band shows the theoretical ±2σ confidence envelope. Watch how drift, volatility, and mean-reversion change the statistics.

SDE MODEL

Process Type Drift μ 0.05 Volatility σ 0.20 Mean-Reversion θ (OU only) 2.0 Paths 12

dX = μdt + σdW

Science: Itô calculus: d(f(X)) = f'dX + ½f''(dX)². GBM solution: X(t) = X₀exp((μ-σ²/2)t + σW(t)). OU is stationary Gaussian with variance σ²/(2θ) — used for interest rates (Vasicek model).