1. Itô’s Isometry: an attempt for derivation

Itô’s isometry is the identity that allows stochastic integration with respect to Brownian motion to be understood as an \(L^2\)-isometric operation.
Rather than being a computational trick, it reflects the orthogonality and variance structure of Brownian increments.

1.1 Framework and statement

Let \((W_t)_{t \ge 0}\) be a standard Brownian motion defined on a filtered probability space
\((\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})\). Furtheremore, let \(H_t\) be an adapted process satisfying the square-integrability condition

\[ \begin{equation} \mathbb{E}\left[\int_0^T H_t^2\,dt\right] < \infty. \label{eq:square_integrability_condition} \end{equation} \]

Then the Itô integral satisfies the identity

\[ \begin{equation} \mathbb{E}\left[\left(\int_0^T H_t\,dW_t\right)^2\right] = \mathbb{E}\left[\int_0^T H_t^2\,dt\right]. \label{eq:ito_isometry} \end{equation} \]

Why 'Isometry'?

The term "isometry" comes from the fact that this formula preserves the \(L^2\) norm — the stochastic integral operator maps \(L^2([s,t])\) to \(L^2(\Omega)\) isometrically, meaning:

\[ \left\|\int_s^t f(u)\,dW(u)\right\|_{L^2(\Omega)} = \|f\|_{L^2([s,t])} \]

This is analogous to how rotations in Euclidean space preserve lengths.

1.2 Reduction to simple adapted processes

The construction of the Itô integral begins with simple adapted processes of the form in \(\ref{eq:simple_process}\).

\[ \begin{equation} H_t = \sum_{k=0}^{n-1} H_k\,\mathbf{1}_{(t_k,t_{k+1}]}(t), \label{eq:simple_process} \end{equation} \]

where

partitions are \(0 = t_0 < t_1 < \dots < t_n = T\), with \(n\) the number of partitions.
each coefficient \(H_k\) is \(\mathcal{F}_{t_k}\)-measurable,
and \(\mathbb{E}[H_k^2] < \infty\).

For such processes, the Itô integral is defined by \(\ref{eq:ito_integral_definition}\), as an aggregation by partitions, where the partition in time \((t_{i+1}-t_i) \to 0\).

\[ \begin{equation} \int_0^T H_t\,dW_t := \sum_{k=0}^{n-1} H_k\left(W_{t_{k+1}} - W_{t_k}\right). \label{eq:ito_integral_definition} \end{equation} \]

1.3 Second moment of the stochastic integral

We now compute the second moment of the integral defined in \(\ref{eq:ito_integral_definition}\), by introducing the Brownian increments in \(\ref{eq:brownian_increment}\) below.

\[ \begin{equation} \Delta W_k := W_{t_{k+1}} - W_{t_k}. \label{eq:brownian_increment} \end{equation} \]

Then, considering the continuous integral can be expressed as summatory when the lenght of partitions tend to zero, we have \(\ref{eq:second_moment_integral}\) below.

\[ \begin{equation} \mathbb{E}\left[\left(\int_0^T H_t\,dW_t\right)^2\right] = \mathbb{E}\left[ \left( \sum_{k=0}^{n-1} H_k \Delta W_k \right)^2 \right]. \label{eq:second_moment_integral} \end{equation} \]

Thus, the right hand of \(\ref{eq:second_moment_integral}\) can be also defined as in \(\ref{eq:second_moment_integral_II}\).

\[ \begin{equation} \mathbb{E}\left[ \left( \sum_{k=0}^{n-1} H_k \Delta W_k \right)^2 \right] = \mathbb{E}\left[ \left( \sum_{k=0}^{n-1} H_k \Delta W_k \right)\times \left( \sum_{l=0}^{n-1} H_l \Delta W_l \right) \right]. \label{eq:second_moment_integral_II} \end{equation} \]

Then, we can expand the square, see \(\ref{eq:square_expansion}\), taking into consideration the commutative property of multiplication, giving rise to multiply twice when \(k < l\). This allows to account for the double multiplication without need of double processing.

\[ \begin{equation} \mathbb{E}\left[ \sum_{k=0}^{n-1} H_k^2 (\Delta W_k)^2 + 2 \sum_{k<\ell} H_k H_\ell \Delta W_k \Delta W_\ell \right]. \label{eq:square_expansion} \end{equation} \]

1.3.1 Vanishing of cross terms

For indices \(k < \ell\), the random variable \(H_k \Delta W_k\) is \(\mathcal{F}_{t_{k+1}}\)-measurable, while \(\Delta W_\ell\) is independent of \(\mathcal{F}_{t_{k+1}}\) and has zero mean. Therefore,

\[ \begin{equation} \mathbb{E}\left[ H_k H_\ell \Delta W_k \Delta W_\ell \right] = 0. \label{eq:cross_terms_vanish} \end{equation} \]

All mixed terms vanish, reflecting the orthogonality of disjoint Brownian increments.

1.4 Contribution of diagonal terms

Brownian motion has independent increments with variance proportional to time, so

\[ \begin{equation} \mathbb{E}\left[(\Delta W_k)^2\right] = t_{k+1} - t_k. \label{eq:increment_variance} \end{equation} \]

Since \(H_k\) is \(\mathcal{F}_{t_k}\)-measurable and independent of \(\Delta W_k\),

\[ \begin{equation} \mathbb{E}\left[ H_k^2 (\Delta W_k)^2 \right] = \mathbb{E}\left[H_k^2\right] \left(t_{k+1} - t_k\right). \label{eq:diagonal_terms} \end{equation} \]

Summing over all intervals gives

\[ \begin{equation} \mathbb{E}\left[\left(\int_0^T H_t\,dW_t\right)^2\right] = \sum_{k=0}^{n-1} \mathbb{E}\left[H_k^2\right] \left(t_{k+1} - t_k\right). \label{eq:sum_representation} \end{equation} \]

1.5 Identification with a time integral

The right-hand side can be written as

\[ \begin{equation} \mathbb{E}\left[ \int_0^T H_t^2\,dt \right], \label{eq:time_integral_representation} \end{equation} \]

which follows directly from the definition of \(H_t\) as a simple process. Thus, for simple adapted processes,

\[ \begin{equation} \mathbb{E}\left[\left(\int_0^T H_t\,dW_t\right)^2\right] = \mathbb{E}\left[\int_0^T H_t^2\,dt\right]. \label{eq:ito_isometry_simple} \end{equation} \]

1.6 Extension to general integrands

Simple adapted processes are dense in \(L^2(\Omega \times [0,T])\). The Itô integral is extended to all processes satisfying \(\eqref{eq:square_integrability_condition}\) by continuity.

Since the identity holds for a dense subclass and both sides are continuous in \(L^2\), it extends uniquely to all admissible integrands, yielding Itô’s isometry \(\eqref{eq:ito_isometry}\).

1.7 Interpretation

Itô’s isometry shows that the stochastic integral preserves \(L^2\)-norms in expectation.
Brownian motion plays the role of an orthogonal noise source, transforming time into variance and enabling stochastic calculus to mirror Hilbert-space geometry.