Itô Isometry¶

1. Concept Definition¶

The Itô isometry is the fundamental identity that governs the second moment of stochastic integrals. Since the Itô integral has mean zero (martingale property), its second moment equals its variance. The identity states: for an adapted, square-integrable process \(\beta(t)\),

In words: the \(L^2\) norm of the stochastic integral equals the \(L^2\) norm of the integrand in the space \(\mathcal{L}^2([0,T])\).

This identity plays the same role for stochastic integrals that Parseval's identity plays for Fourier series: energy is preserved under the transform. Stochastic integration is an isometry between two Hilbert spaces:

Distances in \(\mathcal{L}^2([0,T])\) (measured by \(\mathbb{E}[\int \beta^2\,dt]\)) are preserved as distances in \(L^2(\Omega)\) (measured by \(\mathbb{E}[(\int \beta\,dW)^2]\)). This Hilbert space structure is what makes the \(L^2\)-extension of the integral possible.

2. Explanation¶

Heuristic setup¶

Let \(W_t\) be a standard Brownian motion and let \(\beta(t)\) be a simple adapted process: \(\beta(t) = \beta_i\) on \([t_i, t_{i+1})\). The Itô integral is approximated by

Each term \(\beta_i \Delta W_i\) is the gain from holding \(\beta_i\) units of risk over the interval \([t_i, t_{i+1}]\).

Squaring the sum¶

To understand the size of the integral, examine its second moment. Squaring:

Why cross terms vanish¶

Let \((\mathcal{F}_t)_{t \ge 0}\) be the natural filtration of \(W_t\). Fix \(i < j\). Then \(\beta_j\) is \(\mathcal{F}_{t_j}\)-measurable, and \(\Delta W_j\) is independent of \(\mathcal{F}_{t_j}\) with \(\mathbb{E}[\Delta W_j \mid \mathcal{F}_{t_j}] = 0\). Therefore:

Intuition: \(\Delta W_j\) represents future noise, independent of everything known at time \(t_j\). Conditioning kills the future noise. Brownian increments on disjoint intervals are orthogonal random variables in \(L^2(\Omega)\).

Diagonal terms¶

We are left with:

using independence of \(\beta_i\) (adapted to \(\mathcal{F}_{t_i}\)) from the future increment \(\Delta W_i\), and \(\mathbb{E}[(\Delta W_i)^2] = t_{i+1} - t_i\). As the partition is refined, this Riemann sum converges to:

completing the heuristic derivation.

Connection to the \(L^2\) extension¶

The isometry is not just an interesting identity—it is the mechanism of construction. The rigorous extension of the Itô integral from simple processes to all \(\mathcal{L}^2([0,T])\) proceeds by:

1. Defining the integral for simple processes and proving the isometry directly.
2. Using the isometry to show that approximating sequences of simple processes yield Cauchy sequences in \(L^2(\Omega)\).
3. Defining the general integral as the \(L^2(\Omega)\) limit.

Without the isometry, step 2 fails and the extension is impossible. The isometry is the stochastic analogue of Parseval's identity: energy in equals energy out.

3. Diagram¶

flowchart LR
    A["Brownian increments ΔW_i orthogonal in L²(Ω)"] --> B["Multiply by strategy β_i"]
    B --> C["Square the stochastic sum"]
    C --> D["Cross terms vanish (independence + mean zero)"]
    C --> E["Diagonal terms survive: (ΔW_i)² ≈ Δt_i"]
    E --> F["Riemann sum of β² → ∫ β² dt"]
    D --> G["E[(∫ β dW)²] = E[∫ β² dt]"]
    F --> G

4. Examples¶

Example 1: Deterministic integrand \(\beta(t) = t\)¶

Since \(\beta\) is deterministic, the Itô isometry gives:

A deterministic integrand applied to a Gaussian process gives a Gaussian result, so:

Example 2: Random integrand \(\beta(t) = t W_t\)¶

The integral is not Gaussian since \(\beta(t) = tW_t\) is a random integrand.

Example 3: Numerical verification¶

The following script verifies the Itô isometry by comparing both sides across three integrands.

```python import numpy as np

T = 3.0 n_per_year = 252 N = int(T * n_per_year) dt = T / N sqrt_dt = np.sqrt(dt)

n_paths = 20000 seed = 42 rng = np.random.default_rng(seed)

t = np.linspace(0.0, T, N + 1)

Brownian motion from fair coins (Donsker approximation)¶

coins = rng.choice([-1, 1], size=(n_paths, N)) dB = coins * sqrt_dt

B = np.zeros((n_paths, N + 1)) B[:, 1:] = np.cumsum(dB, axis=1)

def ito_integral(integrand_values, dB): """Cumulative left-endpoint Itô sum.""" dI = integrand_values * dB I = np.zeros((dI.shape[0], dI.shape[1] + 1)) I[:, 1:] = np.cumsum(dI, axis=1) return I

Three integrands¶

H1 = t[:-1] * B[:, :-1] # H_s = s B_s H2 = t[:-1] * (B[:, :-1] ** 2) # H_s = s B_s^2 H3 = -np.cos(B[:, :-1]) # H_s = -cos(B_s)

I1_T = ito_integral(H1, dB)[:, -1] I2_T = ito_integral(H2, dB)[:, -1] I3_T = ito_integral(H3, dB)[:, -1]

def check_ito_isometry(name, H, I_T, dt): lhs = np.mean(I_T ** 2) rhs = np.mean(np.sum(H ** 2 * dt, axis=1)) rel_err = abs(lhs - rhs) / rhs if rhs != 0 else float("nan") print(f"{name}") print(f" LHS E[(∫ H dB)²] = {lhs:.6f}") print(f" RHS E[∫ H² dt] = {rhs:.6f}") print(f" Relative error = {rel_err:.4%}")

check_ito_isometry("H_s = s B_s (theoretical RHS = T⁴/4 = 20.25)", H1, I1_T, dt) check_ito_isometry("H_s = s B_s²", H2, I2_T, dt) check_ito_isometry("H_s = -cos(B_s)", H3, I3_T, dt) ```

Typical output:

text H_s = s B_s (theoretical RHS = T⁴/4 = 20.25) LHS E[(∫ H dB)²] = 20.738498 RHS E[∫ H² dt] = 20.231128 Relative error = 2.51% H_s = s B_s² LHS E[(∫ H dB)²] = 151.042892 RHS E[∫ H² dt] = 145.266226 Relative error = 3.98% H_s = -cos(B_s) LHS E[(∫ H dB)²] = 1.749029 RHS E[∫ H² dt] = 1.744969 Relative error = 0.23%

Both sides agree closely in all three cases. The residual errors (~2–4% for the first two cases, <1% for the third) arise from two sources: discretization bias from the coin-flip approximation to Brownian motion, and Monte Carlo sampling error from using a finite number of paths. Both errors decrease as \(N\) and n_paths increase.

For Case 1, the theoretical value is \(\mathbb{E}[\int_0^T t^2 B_t^2\, dt] = \int_0^T t^3\, dt = T^4/4 = 20.25\), which the Monte Carlo estimates bracket from above and below, confirming the isometry.

5. Summary¶

• The Itô isometry converts the variance of a stochastic integral into an ordinary (Lebesgue) integral of the squared integrand.
• Cross terms vanish because Brownian increments on disjoint intervals are orthogonal random variables in \(L^2(\Omega)\).
• The identity gives stochastic integration a Hilbert space structure: it is an isometry \(\mathcal{L}^2([0,T]) \to L^2(\Omega)\).
• This structure enables the \(L^2\)-extension of the integral from simple processes to all adapted square-integrable processes.
• Nearly every structural result about Itô integrals—existence, uniqueness, continuity, the martingale property—traces back to this theorem.

Exercises¶

Exercise 1. Compute the variance of the Ito integral \(\int_0^2 (3t + 1)\, dW_t\) using the Ito isometry.

Exercise 2. Let \(\beta(t) = e^{-\alpha t}\) for \(\alpha > 0\). Compute

and find the limit as \(T \to \infty\).

Exercise 3. In the proof of the Ito isometry for simple processes, the cross terms \(\mathbb{E}[\beta_i \beta_j \Delta W_i \Delta W_j] = 0\) for \(i < j\). Explain why this argument relies on the left-endpoint evaluation. What would happen if we used right-endpoint evaluation instead?

Exercise 4. Let \(\beta(t) = W_t^2\). Use the Ito isometry to compute

Hint: You will need \(\mathbb{E}[W_t^4] = 3t^2\).

Exercise 5. The Ito isometry extends to the polarized form: \(\mathbb{E}[\int_0^T H\, dW \cdot \int_0^T K\, dW] = \mathbb{E}[\int_0^T HK\, dt]\). Use this to compute the covariance \(\operatorname{Cov}(\int_0^1 t\, dW_t,\; \int_0^1 t^2\, dW_t)\).

Exercise 6. Consider two simple processes \(H^{(n)}\) and \(H^{(m)}\) that approximate the same integrand \(H \in \mathcal{L}^2([0,T])\). Using the Ito isometry, show that

as \(n, m \to \infty\). Explain why this makes the sequence of integrals a Cauchy sequence in \(L^2(\Omega)\).

Exercise 7. The analogy between the Ito isometry and Parseval's identity states that stochastic integration preserves \(L^2\) norms. In Fourier analysis, Parseval's identity says \(\sum_n |c_n|^2 = \frac{1}{2\pi}\int |f|^2\, dx\). Write a brief comparison: what plays the role of the Fourier coefficients \(c_n\) in the stochastic setting, and what plays the role of the \(L^2\) norm of \(f\)?