The Ito integral is built as an isometry from predictable square-integrable processes into square-integrable martingales. Predictability enforces nonanticipation, the isometry supplies completion, and localization extends the construction beyond finite second moments.
The predictable \(L^2\) space
Definition. A simple predictable process on \([0,T]\) has the form \[ H_t=\sum_{k=0}^{n-1}H_k1_{(t_k,t_{k+1}]}(t), \] where \(0=t_0<\cdots<t_n=T\), each \(H_k\) is bounded and \(\mathcal F_{t_k}\)-measurable, and \(H_0\) may also specify the value at zero. Define \[ \int_0^TH_t\,\mathrm{d}B_t = \sum_{k=0}^{n-1} H_k(B_{t_{k+1}}-B_{t_k}). \]
Definition. Let \[ \mathcal H_T^2 = \left\{ H:\ H\text{ predictable and } \mathbb E\int_0^TH_t^2\,\mathrm{d}t<\infty \right\}, \] modulo equality under \(P\otimes\mathrm{d}t\), with norm \[ \|H\|_{\mathcal H_T^2}^2 =\mathbb E\int_0^TH_t^2\,\mathrm{d}t. \]
Lemma. Simple predictable processes are dense in \(\mathcal H_T^2\).
Proof outline. The predictable sigma-field is generated by sets \(A\times(s,t]\) with \(A\in\mathcal F_s\), together with time-zero sets. Indicators of these rectangles generate simple predictable processes. Approximate bounded predictable functions by simple functions using the monotone-class theorem, then truncate a general \(H\) in value and time and apply dominated convergence.
The Ito isometry
Theorem. For simple predictable \(H,G\), \[ \mathbb E\!\left[ \left(\int_0^TH_t\,\mathrm{d}B_t\right) \left(\int_0^TG_t\,\mathrm{d}B_t\right)\right] = \mathbb E\int_0^TH_tG_t\,\mathrm{d}t. \] In particular, \[ \mathbb E\left|\int_0^TH_t\,\mathrm{d}B_t\right|^2 = \mathbb E\int_0^TH_t^2\,\mathrm{d}t. \]
Proof. Refine the partitions so both processes are constant on the same intervals. Expanding the product gives terms \[ \mathbb E[H_iG_j\Delta B_i\Delta B_j]. \] If \(i<j\), everything except \(\Delta B_j\) is \(\mathcal F_{t_j}\)-measurable, so conditional centering makes the expectation zero; similarly for \(j<i\). On the diagonal, \[ \mathbb E[H_iG_i(\Delta B_i)^2] = \mathbb E[H_iG_i]\,(t_{i+1}-t_i), \] because the increment is independent of \(\mathcal F_{t_i}\). Summation proves the identity.
Theorem (construction). There is a unique linear isometry \[ I_T:\mathcal H_T^2\to L^2(\Omega),\qquad H\mapsto\int_0^TH_t\,\mathrm{d}B_t, \] extending the simple-process definition.
Proof. Choose simple predictable \(H^n\to H\) in \(\mathcal H_T^2\). The isometry makes \(I_T(H^n)\) Cauchy in \(L^2(\Omega)\); define \(I_T(H)\) as its limit. If another approximation is used, the isometry applied to the difference shows the same limit. Linearity and preservation of the norm pass to the completion.
The integral process
Theorem. For \(H\in\mathcal H_T^2\), the process \[ M_t=\int_0^tH_s\,\mathrm{d}B_s \] has a continuous modification and is a square-integrable martingale with \[ [M]_t=\langle M\rangle_t =\int_0^tH_s^2\,\mathrm{d} s. \] Moreover, \[ \mathbb E[(M_t-M_s)^2\mid\mathcal F_s] = \mathbb E\!\left[ \int_s^tH_u^2\,\mathrm{d} u \middle|\mathcal F_s\right]. \]
Proof. For simple \(H\), conditional centering of future Brownian increments gives the martingale identity. Expanding the square shows \[ M_t^2-\int_0^tH_s^2\,\mathrm{d} s \] is a martingale. Approximation in \(\mathcal H_T^2\), Doob's \(L^2\) inequality, and the isometry give uniform-in-time convergence in probability of the simple integral processes, so their continuous modifications converge to a continuous modification of \(M\). The martingale and bracket identities pass to the limit. Subtracting at \(s\) and conditioning gives the final formula.
Burkholder-Davis-Gundy inequalities
Theorem. For every \(p>0\), there are constants \(0<c_p\le C_p<\infty\) such that, for every continuous local martingale \(M\) with \(M_0=0\) and stopping time \(\tau\), \[ c_p\,\mathbb E[M]_\tau^{p/2} \le \mathbb E\left(\sup_{t\le\tau}|M_t|^p\right) \le C_p\,\mathbb E[M]_\tau^{p/2}, \] whenever either side is finite.
Proof outline. A good-\(\lambda\) inequality compares the events that the maximum is large while the bracket is small, and conversely. Iterating the inequality and integrating tail probabilities yields equivalence of the two \(L^p\) quantities. Stopping and localization remove boundedness assumptions. For \(p=2\), the upper estimate also follows from Doob's inequality and the Ito isometry.
Corollary. If \(H^n\to H\) in \[ \mathbb E\left(\int_0^T|H_s^n-H_s|^2\,\mathrm{d} s\right)^{p/2}, \] then \[ \mathbb E\sup_{t\le T} \left|\int_0^t(H_s^n-H_s)\,\mathrm{d}B_s\right|^p\to0. \]
Local square integrability
Definition. If \(H\) is predictable and \[ \int_0^tH_s^2\,\mathrm{d} s<\infty \quad\text{almost surely for every }t, \] define \[ \tau_n=\inf\left\{t:\int_0^tH_s^2\,\mathrm{d} s\ge n\right\}. \] The localized integrals agree on overlaps and define the continuous local martingale \[ \int_0^tH_s\,\mathrm{d}B_s. \]
Example. A deterministic \(h\in L^2[0,T]\) gives a centered Gaussian variable \[ \int_0^Th_s\,\mathrm{d}B_s \sim N\!\left(0,\int_0^Th_s^2\,\mathrm{d} s\right). \] A random adapted integrand need not produce a Gaussian variable: \[ \int_0^tB_s\,\mathrm{d}B_s =\frac12(B_t^2-t), \] which is not Gaussian. The isometry determines the second moment, not the entire law.