Continuous Versions of L2 Ito Integrals

From Mean-Square Limits to Continuous Paths

For a progressively measurable integrand \(H\) satisfying \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\), the Ito integral \(\int_0^t H_s\,dB_s\) is initially constructed as an \(L^2(\mathbb{P})\) limit at each fixed time \(t\). Convergence at each fixed time does not, by itself, give a process with continuous paths. The key additional ingredient is a maximal estimate: it controls the largest difference between two approximating integral processes over the whole time interval.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a Brownian motion \(B_t\) relative to the filtration. Fix \(T>0\). All integrands below are progressively measurable and satisfy \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\); this is the class \(\mathcal{L}^2\) on \([0,T]\). By the density of simple predictable integrands, choose simple predictable \(H^n\) with \(\mathbb{E}\int_0^T|H^n_s-H_s|^2\,ds\to0\). Write \(I^n_t=\int_0^t H^n_s\,dB_s\). Each simple integral process has continuous paths almost surely, since it is a finite sum of adapted coefficients multiplied by Brownian increments over time intervals.

The Maximal Estimate for the Approximations

For simple predictable integrands, the Ito integral is a square-integrable martingale: its increments have conditional mean zero given the past, and the Ito isometry gives \(\mathbb{E}|I^n_T|^2=\mathbb{E}\int_0^T|H^n_s|^2\,ds<\infty\). The same facts apply to the difference \(I^n-I^m=\int_0^\cdot(H^n_s-H^m_s)\,dB_s\). Doob’s \(L^2\) maximal inequality, established in the earlier material on maximal bounds, says that a square-integrable martingale \(M\) with continuous paths satisfies \(\mathbb{E}\sup_{0\le t\le T}|M_t|^2\le4\mathbb{E}|M_T|^2\). Combining it with the Ito isometry yields

The supremum in this inequality is measurable: for a continuous path it equals the supremum over a fixed countable dense subset of \([0,T]\) containing \(T\), almost surely. Equivalently, one can first apply the maximal inequality to finitely many times in that subset and then increase the finite sets. This estimate is stronger than a bound at each separate time; it controls the entire path of the difference.

Proof. Density and convergence in \(\mathcal{L}^2\) allow a subsequence satisfying \(\big(\mathbb{E}\int_0^T|H^{n_k}_s-H_s|^2\,ds\big)^{1/2}\le2^{-2k-3}\). The triangle inequality in \(L^2([0,T]\times\Omega)\) gives \(\|H^{n_k}-H^{n_{k+1}}\|_2\le2^{-2k-3}+2^{-2k-5}=5\cdot2^{-2k-5}\). The maximal estimate therefore gives \(\mathbb{E}\sup_t|I^{n_k}_t-I^{n_{k+1}}_t|^2\le4(5\cdot2^{-2k-5})^2=25\cdot2^{-4k-8}\). Markov’s inequality at threshold \(2^{-k}\) gives \(\mathbb{P}(\sup_t|I^{n_k}_t-I^{n_{k+1}}_t|>2^{-k})\le25\cdot2^{-2k-8}\), and the sum of these probabilities is finite. By the Borel–Cantelli lemma, almost surely only finitely many of these events occur. Thus, eventually, each successive uniform difference is at most \(2^{-k}\). Since \(\sum_k2^{-k}\) converges, the triangle inequality shows that the subsequence is uniformly Cauchy almost surely. The supremum can be understood through a countable dense set as above, so the events used in this argument are measurable. \(\square\)

Constructing an Adapted Continuous Modification

The subsequence lemma gives almost-sure uniform convergence, but defining a limit by saying “use it on the event of uniform convergence” requires care if the filtration is not complete. In particular, the event defined using the supremum over all times should not be declared measurable merely by replacing that supremum with one over rational times: the simple integral paths are continuous only almost surely, not on every outcome. Instead, use countable Cauchy events at each time horizon.

Proof. Choose the subsequence from the lemma and denote its integral processes by \(J^k=I^{n_k}\). For each fixed \(t\in[0,T]\), put \(D_t=(\mathbb{Q}\cap[0,t])\cup\{t\}\), a countable dense subset of \([0,t]\) containing \(t\), and define

Each event inside braces belongs to \(\mathcal{F}_t\): the supremum is over countably many \(q\le t\), and \(J^k_q,J^l_q\) are \(\mathcal{F}_q\)-measurable and hence \(\mathcal{F}_t\)-measurable. Countable unions and intersections preserve measurability, so \(A_t\in\mathcal{F}_t\). On \(A_t\), the numerical sequence \(J^k_t\) converges, since \(t\in D_t\). Define \(X_t=\lim_kJ^k_t\) on \(A_t\), and \(X_t=0\) on \(A_t^c\). This defines an \(\mathcal{F}_t\)-measurable random variable for every \(t\), so \(X\) is adapted.

To verify path continuity, take the countable event \(C\) on which every \(J^k\) has a continuous path; \(\mathbb{P}(C)=1\). The subsequence lemma gives probability one to the Cauchy condition on a countable dense subset of \([0,T]\) containing \(T\). On its intersection with \(C\), continuity makes the sequence uniformly Cauchy on the full interval: for any two indices, the continuous function \(|J^k-J^l|\) has the same supremum over that dense set as over \([0,T]\). Hence \(J^k\) converges uniformly there to a continuous function. On this event every \(A_t\) holds, and \(X_t\) agrees for all \(t\) with that uniform limit. Thus \(X\) has continuous paths almost surely.

Finally, for each fixed \(t\), \(\mathbb{P}(A_t)=1\) by the preceding full-interval event, so \(X_t=\lim_kJ^k_t\) almost surely. The Ito isometry gives \(\mathbb{E}|J^k_t-I_t|^2=\mathbb{E}\int_0^t|H^{n_k}_s-H_s|^2\,ds\to0\), hence \(J^k_t\to I_t\) in probability. The same sequence converges almost surely to \(X_t\), and therefore in probability to \(X_t\). Limits in probability are unique up to almost-sure equality: for every \(\varepsilon>0\), \(\mathbb{P}(|X_t-I_t|>\varepsilon)\le\mathbb{P}(|X_t-J^k_t|>\varepsilon/2)+\mathbb{P}(|J^k_t-I_t|>\varepsilon/2)\to0\). Thus \(X_t=I_t\) almost surely for each fixed \(t\), proving that \(X\) is a continuous modification. \(\square\)

Common Pitfall: A Pointwise Bound Is Not a Maximal Bound

A tempting but invalid step is to take the largest of pointwise second-moment bounds and place the supremum inside the expectation. For example, knowing \(\mathbb{E}|M_t|^2\le2\) at each of three times does not imply \(\mathbb{E}\max_t|M_t|^2\le2\). Consider a discrete martingale with \(M_0=0\), \(M_1=1\) or \(-1\) with equal probability, and, conditional on \(M_1=1\), \(M_2=2\) or \(0\) with equal probability; conditional on \(M_1=-1\), take \(M_2=-2\) or \(0\) with equal probability. The conditional mean of \(M_2\) is \(M_1\), so this is a martingale. Here \(\mathbb{E}M_1^2=1\), \(\mathbb{E}M_2^2=2\), but \(\mathbb{E}\max(|M_1|,|M_2|)^2=\tfrac12\cdot4+\tfrac12\cdot1=\tfrac52>2\). The correction is to use Doob’s maximal inequality, which supplies the factor \(4\) for the square-integrable martingales in this construction.

A second important distinction is that the process \(X\) is adapted because its definition uses the measurable event \(A_t\) at time \(t\), not because a possibly nonmeasurable whole-path convergence event has been added to the filtration. Countable dense-time tests are safe here precisely because they are imposed as the definition of \(A_t\); their equivalence to full uniform Cauchyness is used only on the probability-one event where all approximating paths are continuous.