Ito's isometry
/ EE-toh (Itô) /
Ito's isometry is the single identity that makes the whole construction of the stochastic integral against Brownian motion possible. Because Brownian paths have unbounded variation, we cannot define integral H dB pathwise; instead we need a way to control the SIZE of the integral so that we can extend it from a small, explicit class of integrands to a large one by a limiting argument. The isometry provides exactly that control: it says the L^2 norm of the stochastic integral equals the L^2 norm of the integrand against time. It turns an analytic problem (does this limit exist?) into a Hilbert-space problem (is this map an isometry?), and the answer is yes.
Precisely, let H be a simple predictable process (or, after extension, any predictable process with E[ integral_0^T H_s^2 ds ] < infinity), and let I_T(H) = integral_0^T H_s dB_s be its stochastic integral against a standard Brownian motion B. Then E[ ( integral_0^T H_s dB_s )^2 ] = E[ integral_0^T H_s^2 ds ]. In words: the second moment of the integral equals the expected integral of the square of the integrand. The map H -> integral_0^T H dB is therefore a linear isometry from the space L^2 of predictable integrands (with norm given by the right-hand side) into L^2 of the probability space (with norm given by the left-hand side). The proof for simple integrands is a one-line computation: the cross terms vanish because increments of B over disjoint intervals are independent and centered, given the integrand value (which is known at the left endpoint), and the diagonal terms give E[H_{t_k}^2] * (t_{k+1} - t_k) by E[(B_{t_{k+1}} - B_{t_k})^2 | F_{t_k}] = t_{k+1} - t_k.
This is the load-bearing wall of stochastic integration. Since the simple predictable processes are dense in L^2 of integrands, and the integral preserves L^2 distance exactly, the integral extends uniquely and continuously to all of L^2 by completeness of the target Hilbert space — Cauchy sequences of integrands map to Cauchy sequences of integrals, whose limits define the integral of the limit integrand. The crucial hypotheses you cannot drop: the integrand must be PREDICTABLE (evaluated using information strictly before the increment, so the cross-term independence holds) and square-integrable. If you evaluate at the midpoint instead (Stratonovich) the cross terms no longer vanish and the isometry fails in this clean form. There is also a conditional / running version: E[ (integral_0^T H dB)^2 | F_t ] relates to the integral from t, which is what shows the integral is a martingale with the right quadratic variation.
Take H_s = B_s itself and compute E[ ( integral_0^T B_s dB_s )^2 ]. By the isometry this equals E[ integral_0^T B_s^2 ds ] = integral_0^T E[B_s^2] ds = integral_0^T s ds = T^2 / 2. (One can cross-check using Ito's formula, which gives integral_0^T B dB = (B_T^2 - T)/2, whose second moment is indeed T^2/2.)
The isometry computes a second moment of a stochastic integral without ever evaluating the integral pathwise — the whole point of the construction.
The isometry is an L^2 statement and requires the integrand to be predictable and square-integrable; it does NOT hold for the Stratonovich integral, and for non-square-integrable predictable integrands one must use localization to define the integral as a local martingale instead.