Theory
GBM: dW = μW dt + σW dB
Solution: W(T) = W₀ exp((μ-σ²/2)T + σB_T)
Ensemble average (expected value):
⟨W(T)⟩ = W₀ exp(μT) [grows!]
Time-average growth rate:
g_time = μ - σ²/2 [can be negative!]
Ergodic IFF σ = 0 (no noise)
For GBM, ensemble average grows at rate μ, but almost every individual trajectory grows at rate μ−σ²/2. When σ²/2 > μ, the typical trajectory SHRINKS while the average GROWS — driven by rare lucky paths. This is why average investment returns can be positive while most investors lose money. (Ole Peters, LML)