Here is the problem in one sentence: you cannot integrate against a Brownian path the ordinary way. The Riemann–Stieltjes integral \int H\, dg needs the integrator g to have bounded variation — a finite total up-and-down mileage — and we saw on the previous page that a Brownian path has infinite total variation on every interval, however short. Worse, the Riemann sums for \int_0^T H\, dW genuinely depend on where in each slot you sample the integrand: left endpoint, midpoint and right endpoint give three different limits. Classical integration theory doesn't bend here — it breaks.

In 1944, Kiyoshi Itô made a move of striking simplicity: stop trying to fix the pathwise integral, and change the rules twice. First, commit to the left endpoint — always sample the integrand at the start of each slot, before the increment is revealed. Second, take the limit not path by path but in mean square — an L^2 limit over all the randomness at once. Those two choices give one well-defined object, the Itô integral, and essentially all of quantitative finance — option prices, hedging, risk-neutral measures — is built on top of it.

Throughout, (W_t) is a Brownian motion adapted to a filtration (\mathcal{F}_t), and the integrand H is adapted: H_t is \mathcal{F}_t-measurable — known from the information available at time t, with no peeking into the future. This single word, "adapted", is what makes everything below work. In finance it has a blunt translation: the strategy is not allowed to trade on tomorrow's newspaper.

Stage 1 — simple adapted processes (a trading strategy)

Start with the strategies you could actually trade: hold a fixed (random but already-known) position over each of a finite set of time-slots, rebalancing only at the partition times 0 = t_0 < t_1 < \cdots < t_n = T. Formally, a simple adapted process is

where each coefficient H_{t_i} is \mathcal{F}_{t_i}-measurable — the position you set at the start of the slot (t_i, t_{i+1}], knowing only the past. For such a process there is one natural definition of the integral: in each slot, multiply the held position by the Brownian increment over that slot, and add up.

Read this as a trader would, with W standing in for a (driftless) asset price. You hold H_{t_0} shares over the first slot; the price moves by W_{t_1} - W_{t_0}; your gain on that slot is position times price move, H_{t_0}(W_{t_1} - W_{t_0}). Then you rebalance to H_{t_1} shares, the price moves again, and so on. The sum on the right is nothing exotic — it is the total trading P&L of the strategy H. The Itô integral is the profit-and-loss of continuously rebalanced trading; that is why it, and not some other stochastic integral, is the language of mathematical finance.

The crucial feature is that the coefficient is H_{t_i} — the left endpoint of the slot. The position is fixed before the increment \Delta W_i = W_{t_{i+1}} - W_{t_i} is revealed, so H_{t_i} is independent of its own increment. That independence is the engine of every property that follows.

First computation: \int_0^T dW = W_T

Before anything clever, sanity-check the definition on the simplest integrand of all: H \equiv 1 (hold one share, always). This is a simple adapted process for any partition, with every coefficient equal to 1. The defining sum telescopes:

Every intermediate value cancels against its neighbour, leaving

No limit was even needed — the answer is the same for every partition. Financially: buy one share at time 0 and never touch it, and your P&L is exactly the total price move W_T. So far, so reassuringly ordinary. The surprises begin when the integrand itself is random.

The integral has mean zero — line by line

The first dividend of the left-endpoint choice: the integral of any simple adapted process is mean-zero. We prove \mathbb{E}\big[\int_0^T H\, dW\big] = 0 with no skipped steps. (In trading language: against a driftless price, no adapted strategy has positive expected P&L. There is no free lunch to be manufactured by clever rebalancing.)

Step 1 — expectation of the sum is the sum of expectations. Expectation is linear, so push it through the finite sum term by term:

Step 2 — condition each term on the past. Fix a term and use the tower property, conditioning on \mathcal{F}_{t_i} — everything known when the position was set:

Step 3 — pull out the known position. The coefficient H_{t_i} is \mathcal{F}_{t_i}-measurable, so it is a constant inside the inner conditional expectation and factors out:

Step 4 — the future increment has conditional mean zero. The increment \Delta W_i = W_{t_{i+1}} - W_{t_i} is independent of \mathcal{F}_{t_i} and distributed N(0,\, t_{i+1} - t_i), so conditioning changes nothing and its mean is 0:

Step 5 — every term vanishes, so the sum does. Combining Steps 3 and 4, each conditional expectation is H_{t_i}\cdot 0 = 0, hence each term is 0, hence

Notice where adaptedness did the work: in Step 3 we factored out H_{t_i} only because it was already known, and in Step 4 the increment was independent of it. Had we sampled at the right endpoint, H_{t_{i+1}} would be correlated with \Delta W_i and the term would not vanish — the integral would carry a drift.

The famous example: \int_0^T W\, dW

Now the computation everyone remembers. Take the integrand to be the Brownian path itself, H_s = W_s — an adapted process if ever there was one (the strategy "hold as many shares as the current price"). Approximate it by the simple process that samples at the left endpoints, and write out the defining sum over the partition t_i = iT/n:

Step 1 — an exact algebraic identity. For each slot, expand the square of the endpoint value W_{t_{i+1}} = W_{t_i} + \Delta W_i:

Solve this for the summand — the term we actually want:

This is pure algebra — no probability yet, and no approximation. It holds term by term, exactly.

Step 2 — sum, and let the first part telescope. Summing over i = 0, \dots, n-1, the squared terms cancel in pairs exactly as in our first computation, leaving only the two ends:

Step 3 — the sum of squared increments does not vanish. Here is the moment classical intuition fails. For a smooth path, squaring the (tiny) increments makes them doubly tiny and the sum \sum (\Delta x_i)^2 \to 0. For Brownian motion, each (\Delta W_i)^2 has mean t_{i+1} - t_i — the increments are of size \sqrt{\Delta t}, so their squares are of size \Delta t, and there are 1/\Delta t of them. The sum converges (in mean square) not to zero but to the length of the interval:

This limit is the quadratic variation of Brownian motion — the same [W]_T = T that broke classical calculus on the previous page, now reappearing inside the new integral as an honest, finite contribution.

Step 4 — pass to the limit. Putting Steps 2 and 3 together:

Compare with what first-year calculus would predict. For an ordinary differentiable function, \int_0^T x\, dx = T^2/2 — that is, \int_0^T f\, df = f(T)^2/2 when f(0)=0, with no correction, because the squared-increment sum dies in the limit. The Itô answer keeps the familiar W_T^2/2 but subtracts T/2. That extra term is the fingerprint of quadratic variation: a deterministic drift generated purely by the roughness of the path. Every strange-looking formula in stochastic calculus — Itô's lemma, the \tfrac{1}{2}\sigma^2 in Black–Scholes — is this same -T/2 phenomenon wearing different clothes.

One quick consistency check: the new formula still respects the mean-zero property. Since \mathbb{E}[W_T^2] = T,

exactly as the theorem demands. The correction term is not a bug — it is precisely what keeps the fair game fair.

Stage 2 — general adapted integrands, by approximation

Real integrands are not step functions — the W\,dW example already forced us to approximate. The extension is the standard analyst's move: approximate a general adapted process by simple ones and pass to a limit. Let H be adapted and square-integrable,

One can choose simple adapted processes H^{(m)} that approximate H in this L^2(dt \times d\mathbb{P}) sense,

and define the Itô integral of H as the limit of the simple integrals,

For this to be a sound definition the simple integrals must form a Cauchy sequence in L^2(\Omega), and the limit must not depend on which approximating sequence we picked. Both are guaranteed by a single remarkable identity — the Itô isometry, the subject of the next page — which converts the size of the integral into the size of the integrand, so an approximation that is close in the integrand sense produces integrals that are close in L^2(\Omega). The two-stage scaffolding (define on a dense simple class, extend by isometry) is the same construction that builds the Lebesgue integral and the Fourier transform on L^2 — Itô's originality was seeing that adaptedness plus independence of increments supplies exactly the isometry the extension needs.

Watch the P&L accumulate

Below, a Brownian path (the faint curve) drives a simple step strategy H — a piecewise-constant position, set fresh at the start of each slot and held until the next rebalancing time. The bars show the held position H_{t_i} in each slot; the bold stepped curve traces the running integral \sum_{j \le i} H_{t_j}\,\Delta W_j — the left-endpoint position times the increment, summed slot by slot. This is Stage 1 made visible: the running integral is literally the strategy's cumulative trading P&L.

Two things to watch for as you Refresh. First, whenever the position is 0, the P&L curve goes flat — no position, no exposure, no gain or loss, however wildly the path moves. Second, over many refreshes the final value \int H\, dW lands above and below zero with no systematic preference: you are watching the mean-zero theorem in action. A large final value happens — variance is real — but no strategy in this family can make the average outcome positive.