the Ito integral as a martingale
/ EE-toh /
One of the deepest payoffs of using the left-endpoint (predictable) convention is that the resulting Ito integral is itself a martingale. This is not a cosmetic feature: it means the entire toolkit of martingale theory — optional stopping, the maximal and Doob inequalities, the Burkholder-Davis-Gundy inequalities, convergence theorems — applies to stochastic integrals for free. It is the reason the Ito integral is the natural object of stochastic analysis and why 'integral H dB' models a self-financing trading strategy whose value is a fair game in the no-arbitrage world.
Precisely: if H is predictable with E[ integral_0^T H_s^2 ds ] < infinity, then the process M_t = integral_0^t H_s dB_s is a continuous, square-integrable martingale with M_0 = 0, mean zero (E[M_t] = 0), and quadratic variation [M]_t = integral_0^t H_s^2 ds. The martingale property E[M_t | F_s] = M_s for s <= t is inherited from the construction: on simple integrands the increment of M over (s, t] is a sum of terms xi_k (B_{t_{k+1}} - B_{t_k}) where the height is F_{t_k}-measurable and the Brownian increment is centered and independent given F_{t_k}, so each term has conditional mean zero; the property survives the L^2 limit because conditional expectation is L^2-continuous. The companion fact M_t^2 - integral_0^t H_s^2 ds is also a martingale identifies the quadratic variation and is the running form of the Ito isometry (take expectations to recover E[M_t^2] = E[integral_0^t H^2 ds]).
Where it bites and where the caveats live. Combined with Doob's maximal inequality you get E[ sup_{s <= T} M_s^2 ] <= 4 E[M_T^2] = 4 E[integral_0^T H^2 ds], a basic estimate behind SDE existence proofs. With optional stopping you can evaluate stochastic integrals at stopping times. The honest and important caveat: the TRUE martingale property requires the L^2 (or at least the integrability) condition. If H is only locally square-integrable, integral H dB is merely a continuous LOCAL martingale, and local martingales can fail to be martingales — a strict local martingale has E[M_t] strictly decreasing in t even though M_0 = 0 would suggest a fair game. So 'the Ito integral is a martingale' is true exactly when the integrand is square-integrable in expectation; otherwise replace 'martingale' by 'local martingale' and check Novikov-type or uniform-integrability conditions before quoting mean-zero or optional stopping.
The integral M_t = integral_0^t B_s dB_s = (B_t^2 - t)/2 is a martingale: indeed E[(B_t^2 - t)/2] = (t - t)/2 = 0 for all t, and (B_t^2 - t)/2 is exactly the centered martingale whose quadratic variation is integral_0^t B_s^2 ds. By contrast, a famous strict local martingale, the inverse three-dimensional Bessel process 1/|W_t| (W a 3D Brownian motion started away from 0), is a local martingale but E[1/|W_t|] strictly DECREASES, so it is not a true martingale.
Square-integrable integrand: a true martingale. The 1/|W| example shows local martingales can secretly leak expectation.
Do not assume E[integral H dB] = 0 automatically — that requires the true martingale property (the L^2 or UI condition). For a strict local martingale the expectation can be strictly negative even though the integral starts at 0.