Itô isometry: noise energy equals time energy | QuantXYZ

The Itô isometry says a stochastic integral has the same L2 energy as its integrand against time: E[(∫_0^T H_s dW_s)^2] = E[∫_0^T H_s^2 ds] We prove it from a two-step bet (H = 1, then H = 2), then for simple processes, then by an L2 limit. Independent Brownian increments are orthogonal, so slice energies add. That is Pythagoras for Itô integration.