Exact solution: X(t) = X₀e^{−θt} + μ(1−e^{−θt}) + σ∫₀ᵗ e^{−θ(t−s)}dW(s)
Stationary distribution: N(μ, σ²/2θ) — shown as the histogram below.
Autocorrelation: E[X(t)X(t+τ)] = (σ²/2θ)·e^{−θ|τ|} — exponential decay with rate θ.
Applications: Vasicek interest rate model, velocity in Brownian particle, neuroscience (leaky integrate-and-fire), mean-reverting spreads in pairs trading.
Stationary distribution: N(μ, σ²/2θ) — shown as the histogram below.
Autocorrelation: E[X(t)X(t+τ)] = (σ²/2θ)·e^{−θ|τ|} — exponential decay with rate θ.
Applications: Vasicek interest rate model, velocity in Brownian particle, neuroscience (leaky integrate-and-fire), mean-reverting spreads in pairs trading.