the integral against a continuous local martingale
Brownian motion is only the first integrator. Stochastic analysis really needs to integrate against any continuous local martingale M, because the integrals you build (and the solutions of SDEs) are themselves local martingales, and you want to integrate against them in turn. The construction generalizes the Brownian case almost verbatim, with the single change that the clock dt is replaced by the increasing process d[M], the quadratic variation of M. This is the step that makes the theory closed and self-contained.
The recipe is the same three-step extension. For M a continuous local martingale with quadratic variation [M], and H a predictable integrand, the integral integral_0^t H dM is defined first on simple integrands by the finite sum sum xi_k (M_{t_{k+1}} - M_{t_k}); it satisfies the generalized Ito isometry E[ (integral_0^T H dM)^2 ] = E[ integral_0^T H_s^2 d[M]_s ], where the relevant L^2 space is now L^2(d[M] x dP) — predictable processes H with E[ integral_0^T H^2 d[M] ] < infinity. Density of simple integrands plus completeness extends the integral to that whole space, and localization (stopping when integral H^2 d[M] reaches n) extends it further to all predictable H with integral_0^T H^2 d[M] < infinity almost surely. The resulting process N_t = integral_0^t H dM is again a continuous local martingale with N_0 = 0 and quadratic variation [N]_t = integral_0^t H_s^2 d[M]_s; more generally [integral H dM, integral K dM'] = integral H K d[M, M']. The whole calculus — Ito's formula, the product rule, BDG inequalities — runs with [M] in place of t.
This is the bridge from 'integrate against Brownian motion' to 'integrate against anything that arises'. A continuous semimartingale is, by definition, a continuous local martingale plus a continuous bounded-variation process; integrating against the local-martingale part by this construction and against the bounded-variation part by ordinary Lebesgue-Stieltjes integration combines into the full semimartingale integral. The honest caveats: (1) The output is a LOCAL martingale, not necessarily a true martingale — exactly as in the Brownian case, square-integrability of H against d[M] in EXPECTATION is what upgrades it. (2) The integrand must be predictable and integrable against d[M]; the relevant norm uses [M], not Lebesgue time, so an integrand can be fine for one martingale and not another. (3) Because M may itself be a strict local martingale, even with bounded H the integral can be a strict local martingale, so 'integral H dM is a martingale' is a statement requiring proof (a localizing sequence plus uniform integrability), not a given.
Let M be a continuous local martingale with [M]_t = t (which, by Levy's characterization, is a Brownian motion). Take H_s = 1_{(a, b]}(s). Then integral_0^t H dM = M_{t and b} - M_{t and a}, with quadratic variation integral_0^t H^2 d[M] = (b and t) - (a and t) — the same time-as-clock behaviour as in the Brownian case, now justified through [M] rather than assumed. For a general M, replacing [M]_t = t by a strictly increasing [M] simply reparametrizes the clock against which the integral accumulates variation.
Integration against M is integration against Brownian motion with the clock dt replaced by d[M].
The L^2 norm of the integrand is taken against d[M], not Lebesgue time, and the output is a local martingale; promoting it to a true martingale still needs E[integral H^2 d[M]] < infinity (or a uniform-integrability argument), because M itself may be a strict local martingale.