Itô Isometry for Stochastic Integrals Explained
Learn how Itô isometry turns the variance of a Brownian stochastic integral into an ordinary time integral and why that matters for hedging and simulation
Direct answer
Itô isometry says that the expected square of a square-integrable Brownian stochastic integral equals the expected time integral of its squared integrand. It converts random path risk into an ordinary variance calculation, but only after predictability and integrability are stated
The isometry measures squared uncertainty
For a predictable process H, the central identity is E[(∫₀ᵀ H_t dW_t)²] = E[∫₀ᵀ H_t² dt]
The integral has conditional mean zero when its assumptions hold, so the two sides also describe its variance
This is not a rule for multiplying differentials. It is an L² identity defined through limits of simple trading-time approximations
Step processes show why the squares add
Suppose H equals 2 from time 0 to 0.25 and zero afterward. Then the integral is 2(W_0.25 - W_0)
Its variance is 4 × 0.25 = 1, exactly matching the time integral of H²
For several non-overlapping intervals, independent Brownian increments make cross terms vanish in expectation. The limit preserves that accounting for admissible H
Hedging error needs an exposure path
In a diffusion hedge, a residual exposure H_t to a Brownian shock accumulates as ∫ H_t dW_t
Itô isometry says the modeled mean-squared hedging error depends on the full exposure path, not only its final delta or one average volatility
Two hedges can have the same terminal error on one realized path and very different expected squared errors because their intraday exposure differs
Simulation uses the same variance budget
Monte Carlo increments usually take the form H_t√Δt Z, where Z has mean zero and variance one
Their conditional variance is H_t²Δt. Summing those terms is the discrete counterpart of the isometry
If a simulation changes the time grid, compare the integrated H² budget and sampling error rather than assuming that more paths alone validate a hedge result
Correlation, jumps, and bad inputs remain separate
With several Brownian drivers, the variance includes the covariance matrix and cross-exposure terms
Jump risk, discrete rebalancing, transaction costs, stale quotes, and parameter error are not erased by the identity. They require their own model terms or stress cases
The theorem checks a square-integrable diffusion component. It does not forecast profit, establish market completeness, or make a trading rule safe
Common questions
What does Itô isometry say in simple terms?
It says that the variance created by many small Brownian shocks can be computed by adding the squared size of the exposure over time
Does Itô isometry mean a hedge has zero risk?
No. It only describes the square-integrable Brownian part under a stated model. Jumps, discrete hedging, costs, and incorrect inputs can still create losses
Why is predictability required?
The integrand must be chosen using information available before the Brownian increment. Otherwise it could use future noise and the stochastic integral would not represent an admissible exposure