The Itô Isometry and Martingale Property of Stochastic Integrals

The variance rule

\mathbb{E}[(\int_0^T H_s \, dW_s)^2] = \mathbb{E}[\int_0^T H_s^2 \, ds]

is not merely a convenient formula — it is the mathematical foundation on which the entire stochastic integral is built. It says that the map

H \mapsto \int_0^T H \, dW

is an isometry from the Hilbert space

L^2(\Omega \times [0,T])

(the space of square-integrable adapted processes) into

L^2(\Omega)

(the space of square-integrable random variables). This is why the integral of bounded continuous adapted processes can be extended, via density, to all of

L^2(\Omega \times [0,T])

The martingale property says more: not only is $\int_0^T H \, dW$ in $L^2(\Omega)$ , but the whole process $Z_t = \int_0^t H_s \, dW_s$ is a martingale. This is the mathematical form of the no-arbitrage principle: a self-financing strategy in a BM-driven market generates a martingale P&L under the risk-neutral measure.

Learning objectives. By the end of this note you should be able to:

State the Itô isometry precisely and explain why it is an isometry between Hilbert spaces.
Use the isometry to compute $\mathbb{E}[Z_t^2]$ , $\text{Cov}(Z_s, Z_t)$ , and $\mathbb{E}[Z_s Z_t]$ for specific integrands.
Prove the martingale property of $Z_t = \int_0^t H_s \, dW_s$ for square-integrable $H$ .
Distinguish between a square-integrable martingale and a local martingale; name a sufficient condition for the former.
Connect the isometry to the concept of mean-squared hedging error in derivative pricing.

In this lesson

The Itô Isometry
Covariance Structure of Stochastic Integrals
The Martingale Property
Connection to Derivative Pricing

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