Itô’s isometry is a fundamental result in stochastic calculus, specifically within the context of Itô integrals, which are integrals defined with respect to Brownian motion or more general martingales. Itô’s isometry provides a way to compute the expected value of the square of an Itô integral. This result is key in simplifying the analysis of stochastic integrals, making it useful for theoretical purposes and practical computations in fields like mathematical finance, physics, and engineering.

1. Statement of Itô’s Isometry

Suppose $W_t$ is a standard Brownian motion and $f(t, \omega)$ is a square-integrable stochastic process adapted to the Brownian motion filtration (i.e., $f$ is predictable). Then, for the Itô integral:

$I = \int_0^T f(s) \, dW_s,$

Itô’s isometry states that:

$\mathbb{E} \left[ \left( \int_0^T f(s) \, dW_s \right)^2 \right] = \mathbb{E} \left[ \int_0^T f(s)^2 \, ds \right].$

In words, the expected value of the square of the Itô integral (the left-hand side) is equal to the expected value of the integral of the square of the integrand $f(s)$ over the time interval $[0, T]$ (the right-hand side).

2. Intuition and Importance

• Intuition: Itô’s isometry shows that the “size” or “variance” of an Itô integral can be measured by the squared integrand $f(s)^2$ . It effectively treats the Itô integral as if it behaves similarly to an ordinary $L^2$ -integral, but with respect to Brownian motion as the integrator.
• Interpretation: This formula allows us to measure the mean square of an Itô integral, providing a way to quantify the variability introduced by the stochastic process $f(s)$ in the context of Brownian motion $W_s$.

3. Why Itô’s Isometry Is Useful

• Simplification: Itô’s isometry simplifies calculations involving Itô integrals, particularly when working with the variance or expected square of the integral.
• Foundation for $L^2$ -Space Theory: Itô’s isometry demonstrates that the Itô integral is a bounded operator on $L^2$ -spaces. This property is useful for defining and analyzing stochastic processes in $L^2$ .
• Martingale Property: Itô’s isometry is foundational in proving that the Itô integral of a predictable process with respect to Brownian motion is a martingale.

4. Example Application of Itô’s Isometry

Suppose $f(s) = 1$ is a constant function, and we consider the integral of $f(s)$ with respect to Brownian motion $W_t$ :

$I = \int_0^T 1 \, dW_s = W_T.$

Using Itô’s isometry:

$\mathbb{E} \left[ \left( \int_0^T 1 \, dW_s \right)^2 \right] = \mathbb{E} \left[ \int_0^T 1^2 \, ds \right] = \mathbb{E}[T] = T.$

This result is consistent with the fact that $W_T$ has variance $T$ , as $W_T \sim \mathcal{N}(0, T)$ for standard Brownian motion.

5. Extensions and Generalizations

• Multiple Integrals: Itô’s isometry can be extended to multiple integrals and more complex settings, such as integrals involving martingales other than Brownian motion.
• Applications to Itô’s Lemma: Itô’s isometry is often used in proofs involving Itô’s Lemma, especially when calculating variances and expectations of terms that arise in stochastic differential equations (SDEs).
• Stochastic Differential Equations: When solving SDEs, Itô’s isometry helps in evaluating the second moment of the solutions and analyzing the stability of solutions in $L^2$ -spaces.

6. Formal Proof Outline (Sketch)

The proof of Itô’s isometry relies on approximating the Itô integral by a sequence of simple processes. A simple process $\phi(s)$ might take the form of a step function:

$\phi(s) = \sum_{i=0}^{n-1} \phi_i \mathbf{1}{[t_i, t{i+1})}(s),$

where $\phi_i$ are constant values on intervals $[t_i, t_{i+1})$ . Using properties of Brownian increments and taking the limit as the partition becomes finer, one can rigorously show that:

$\mathbb{E} \left[ \left( \int_0^T \phi(s) \, dW_s \right)^2 \right] = \mathbb{E} \left[ \int_0^T \phi(s)^2 \, ds \right].$

By then extending this result to general square-integrable adapted processes $f(s)$ , the full isometry is established.

7. Implications for Stochastic Analysis

Itô’s isometry is central to developing the $L^2$ -theory of stochastic integration, proving existence and uniqueness results for SDEs, and ensuring that certain stochastic processes have well-defined statistical properties. It also underpins much of the mathematical rigor in stochastic process theory, where expectations, variances, and higher moments are crucial for understanding process behavior over time.