Constructing the Integral by L2 Limits

From Simple Sums to General Integrands

The Ito integral is first defined for simple predictable processes: on each time interval, the coefficient is known at the interval's left endpoint, before the Brownian increment is observed. A general square-integrable integrand need not be constant on any interval, so its integral is constructed by approximation. The key point is that the Ito isometry controls the mean-square difference of two simple integrals by the mean-square difference of their integrands.

Fix \(T>0\) and a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a Brownian motion \(B_t\) adapted to the filtration, with increments independent of the past. In this tutorial every stochastic integral is an Ito integral with respect to this \(B_t\). The admissible integrands are progressively measurable processes \(H_s\) satisfying \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\); this class is denoted by \(\mathcal{L}^2\). Processes that agree outside a set of zero measure for the product measure \(d\mathbb{P}\,ds\) represent the same integrand.

The earlier results on simple predictable processes give the Ito isometry: if \(G\) is simple predictable, then \(\mathbb{E}\big[(\int_0^T G_s\,dB_s)^2\big]=\mathbb{E}\int_0^T G_s^2\,ds\). We will use this identity only for simple processes. We also use the density of simple predictable processes in \(\mathcal{L}^2\), established in the earlier material on that density. These two facts are enough to construct a limit; no pathwise Riemann-sum convergence is required.

The L2-Limit Construction

Proof. We must show that the integrals of the chosen approximants converge in \(L^2(\mathbb{P})\). For any \(m,n\), \(H^n-H^m\) is again simple predictable: refine the two finite partitions to their common partition, and on each refined interval subtract the two coefficients known at its left endpoint. Linearity of the simple integral and the Ito isometry therefore give

For real numbers \(x,y\), \(|x-y|^2\le 2|x|^2+2|y|^2\). Apply this with \(x=H^n_s-H_s\) and \(y=H^m_s-H_s\), then integrate and take expectations. The right-hand side is at most \(2\mathbb{E}\int_0^T|H^n_s-H_s|^2ds+2\mathbb{E}\int_0^T|H^m_s-H_s|^2ds\), which tends to zero as \(m,n\to\infty\). Thus the simple integrals form a Cauchy sequence in \(L^2(\mathbb{P})\). Since \(L^2(\mathbb{P})\) is complete, there is an \(L^2\) random-variable equivalence class to which they converge. This defines the claimed limit for the chosen sequence. \(\square\)

The theorem describes a construction from a selected approximating sequence. The assertion that the limit is unchanged if a different sequence is selected is a separate well-definedness property; it is not needed to show that this particular sequence has a limit. At a time \(t\in[0,T]\), the same construction applies on \([0,t]\): restrict the integrands to that interval, splitting a simple process's interval at \(t\) if necessary, and use the same argument.

In practice, the approximation is chosen to respect the information available at each time. For a process that varies with time, left-endpoint values are often useful: they are measurable at the start of the interval and can approximate the integrand in the \(\mathcal{L}^2\) norm. The norm, rather than pointwise closeness alone, is what the construction requires.

Worked Approximations

Why Predictability Matters

The approximation method is not simply “replace the process by nearby values.” Coefficients must be known at the beginning of the interval whose Brownian increment they multiply. This is the predictable structure that makes the simple Ito integral and its isometry applicable. In the example \(H_s=B_s\), replacing the left endpoint \(B_{k/n}\) by the right endpoint \(B_{(k+1)/n}\) gives coefficients that depend on the increment being integrated over. That is not an admissible simple predictable approximation.

For a concrete illustration, on the single interval \((0,1]\) the right-endpoint choice would give coefficient \(B_1\). It is not \(\mathcal{F}_0\)-measurable (Brownian motion starts at \(B_0=0\), while \(B_1\) is nonconstant). Treating the resulting product as an Ito simple integral would give \(B_1(B_1-B_0)=B_1^2\), whose expectation is \(1\). By contrast, the admissible left-endpoint coefficient is \(B_0=0\), giving the simple integral \(0\) on this one-interval partition. The right-endpoint calculation is not a competing construction: it violates the predictability requirement and cannot be justified by the simple-process Ito isometry.

A second common mistake is to require the approximants to be pointwise close at every time. The construction only requires convergence in \(\mathbb{E}\int_0^T|H^n_s-H_s|^2ds\). This allows random errors and does not require uniform pathwise approximation. Once this norm tends to zero, the isometry proves the integral sequence is Cauchy; completeness of \(L^2(\mathbb{P})\) supplies its limit.