the construction of the Ito integral
/ EE-toh /
The construction of the Ito integral answers the question Vol I could only sketch: given a Brownian motion B and a predictable, square-integrable process H, what is integral_0^T H_s dB_s, when the integrator has paths of unbounded variation that forbid any pathwise Riemann-Stieltjes definition? The answer is an L^2 limit, not a pathwise limit, and the whole construction is a clean instance of the 'define on a dense subspace, then extend by continuity' strategy from functional analysis, with the Ito isometry supplying the continuity.
The construction proceeds in three movements. First, on the simple predictable integrands, define the integral by the explicit finite sum integral_0^T H dB = sum_k xi_k (B_{t_{k+1}} - B_{t_k}); this is unambiguous and requires no limit. Second, verify the Ito isometry on this class: E[ (integral_0^T H dB)^2 ] = E[ integral_0^T H_s^2 ds ], so the integral is a linear isometry from the simple integrands into L^2 of the probability space. Third, invoke density and completeness: the simple predictable processes are dense in the Hilbert space H^2 of predictable processes with E[integral_0^T H^2 ds] < infinity, and L^2 of the probability space is complete; an isometry on a dense subspace of a metric space extends uniquely to a continuous (in fact isometric) map on the whole space. Concretely, given any H in H^2 pick simple H^(n) -> H in H^2; the isometry makes integral H^(n) dB a Cauchy sequence in L^2, its limit is by definition integral H dB, and the limit does not depend on the approximating sequence. One then defines the integral as a PROCESS, integral_0^t H dB for all t, and verifies it can be chosen continuous in t.
Beyond the square-integrable case, one extends further by localization: if H is only predictable with integral_0^T H_s^2 ds < infinity almost surely (no expectation finiteness), one stops at times tau_n where the running integral of H^2 first reaches n, applies the L^2 theory on each piece, and stitches the results; the outcome is a stochastic integral that is a continuous LOCAL martingale rather than a true martingale. The properties earned by the construction are exactly the ones used everywhere: linearity in H, the integral is a continuous (local) martingale started at 0, the Ito isometry (in the L^2 case), and the quadratic variation [integral H dB]_t = integral_0^t H_s^2 ds. The honest caveat: the Ito integral is defined as an equivalence class up to indistinguishability, the convergence is in L^2 (or in probability after localization) and NOT pathwise, and the predictability of H is essential at every step — these are not the same object you would get from naive pathwise Stieltjes integration, which does not exist here.
To define integral_0^T B_s dB_s, approximate the continuous integrand B by the simple integrand H^(n)_s = B_{t_k} on (t_k, t_{k+1}] over a mesh of size T/n. Then integral H^(n) dB = sum B_{t_k}(B_{t_{k+1}} - B_{t_k}), which a short algebraic identity rewrites as (B_T^2 - sum (B_{t_{k+1}} - B_{t_k})^2)/2. As the mesh shrinks the sum of squared increments converges in L^2 to T, so the L^2 limit is integral_0^T B dB = (B_T^2 - T)/2 — the extra -T/2 is the fingerprint of the unbounded variation and the left-endpoint convention.
The construction delivers integral_0^T B dB = (B_T^2 - T)/2; ordinary calculus would have predicted B_T^2 / 2, and the missing -T/2 is the Ito correction.
The integral process can be (and is, by convention) chosen continuous in t and is a martingale only under the L^2 condition E[integral H^2 ds] < infinity; under the weaker pathwise condition it is merely a continuous local martingale, which need not be a true martingale.