Ito Integral

The consistent way to accumulate noise: sum past-only heights times forward Brownian steps. Mean zero, variance equal to the integrated squared height.

Left-point heights times forward Brownian steps sum to mean zero with variance 12.0 for height 2.0 over 3 units

Why Does This Exist?

Riemann sums need smooth paths: heights times widths converge as meshes shrink. Brownian paths have infinite first variation, so those sums oscillate forever and the limit depends on where in each interval you sample the height. Left, middle and right choices give different answers, and stochastic models cannot tolerate that ambiguity.

The Ito integral fixes the evaluation at the left point, keeping integrands non-anticipating (past-only). That choice yields the martingale property (fair game, mean zero) and the isometry (variance equals integrated squared height), the two workhorses of SDE analysis. Stochastic calculus and SDEs frames the toolkit; Itos lemma is the chain rule its integrals obey.

Prerequisite in one paragraph. You need Brownian increments $W_{t+\Delta} - W_t$ with mean $0$ and variance $\Delta$ , independent of the past. Adapted means $H_t$ depends only on history to $t$ . Background is in Brownian motion and integrals definite and indefinite.

Think of It Like This

Betting slips filled before each race

A gambler bets on 100 horse races in sequence, sizing each bet from past results only. The total profit is sum of bet size times race outcome. Because bets never peek at the upcoming race, the game is fair and the variance is sum of squared bet sizes times race variance.

Ito integration is that ledger. Height $H_t$ is the bet size set before the step; Brownian increment is the race payoff. Stratonovich instead lets the gambler split the difference with insider knowledge of the finish, adding a drift that breaks fairness.

Where the analogy stops: race odds stay fixed while Brownian variance scales with $dt$ . The ledger also passes to continuous time, where bets adjust every instant.

How It Actually Works

For a step integrand $H$ constant at $h_i$ on $[t_i, t_{i+1})$ , define $\int_0^T H\,dW = \sum h_i(W_{t_{i+1}} - W_{t_i})$ , where the height uses the left endpoint $t_i$ . Extend to adapted square-integrable $H$ by $L^2$ limits. Then $E[\int_0^T H\,dW] = 0$ (mean zero) and $E[(\int_0^T H\,dW)^2] = E[\int_0^T H_t^2\,dt]$ (Ito isometry: variance equals expected integrated square).

Worked example

Take constant height $H_t = 2.0$ over $T = 3$ . The integral is $2.0 \times W_3$ , a scaled Brownian value. Mean $0$ exactly. Variance $2.0^2 \times 3 = 12.0$ by the isometry, so standard deviation $\sqrt{12} \approx 3.46$ . A simulated value of $4.1$ sits $1.19$ standard deviations out, unremarkable.

Contrast the peeking choice $H_t = W_T$ (future value as height): the sum gains mean $T = 3$ instead of $0$ . Same path, different rule, different answer. The left-point discipline is load-bearing, not cosmetic.

Watch Out For

Coding a midpoint rule and calling it Ito

An Euler scheme sampling $H$ at interval midpoints converges to Stratonovich, adding $\frac{1}{2}\int H' \sigma\,dt$ drift. Diffusion samplers coded that way target a tilted density while diagnostics report the intended one.

Fix: left endpoints for Ito SDEs. State the convention beside every discretisation.

Pulling random heights outside the expectation

The isometry needs $E[\int H^2\,dt]$ with the expectation inside for random $H$ . Replacing it by $(\int E|H|\,dt)^2$ or dropping the square understates variance by Jensen's gap and breaks every downstream confidence band.

Fix: square first, integrate, then expect. Write all three steps explicitly.

The Quick Version

Ito sums left-point heights times forward Brownian steps for past-only integrands.
Mean zero always; variance equals expected integrated squared height.
Constant 2.0 over 3 units gives variance 12.0 and standard deviation 3.46.
Midpoints give Stratonovich with extra drift, not Ito.
Square before integrating and expecting, never after.