Fix an adapted square-integrable integrand H and let the upper limit run, turning the Itô integral into a process:

This object is the workhorse of mathematical finance — a self-financing trading gain, a risk-neutral price increment — and it inherits a distinct personality from its construction. Like every integral it is linear, so portfolios decompose into legs. But the left-endpoint construction against Brownian increments gives it two gifts no ordinary integral has: it is a continuous martingale — zero mean, no drift, a fair bet on a fair game stays fair — and its variance is exactly computable via the Itô isometry.

Linearity, the zero-mean martingale property, and the computable variance: these three do perhaps 90% of the daily work in quantitative finance. Split a book into positions — linearity. Argue that an unhedged noise exposure earns nothing on average — the martingale property. Put a number on the risk of that exposure — the isometry. This page collects the full list of properties, proves the martingale one line by line, and then works each of the three big ones on a concrete example.

The martingale property, line by line

We show \mathbb{E}[M_t \mid \mathcal{F}_s] = M_s for all s \le t. The whole proof rests on one structural fact: the Itô integral, like an ordinary integral, is additive over adjacent intervals.

Step 1 — split the integral at the present time s. The integral from 0 to t is the integral up to s plus the integral over the remaining slice (s, t]:

Step 2 — take the conditional expectation and use linearity. Condition on everything known at s and split the two pieces:

Step 3 — the past term is known. The running integral M_s is \mathcal{F}_s-measurable — it is built only from increments up to time s — so it passes through the conditioning untouched:

Step 4 — the future slice has conditional mean zero. The increment \int_s^t H\, dW is itself an Itô integral, built from left-endpoint positions H_{t_i} (each known at t_i \ge s) times future increments that are independent of \mathcal{F}_s and mean-zero. The very argument that gave the integral mean zero, applied conditionally on \mathcal{F}_s, gives

Step 5 — collect. Adding Steps 3 and 4,

So M_t = \int_0^t H\, dW is a martingale: the best forecast of any future value of the integral, given the present, is its current value. "Martingale in, martingale out" — integrating an adapted bet against Brownian motion never creates drift. Taking plain expectations at s = 0 (where M_0 = 0) gives the zero-mean property as a corollary: \mathbb{E}[M_t] = \mathbb{E}[M_0] = 0 for every t.

Worked example 1 — linearity: split the book into legs

Linearity is the least glamorous property and the one used most often. A portfolio's gain integral splits into the gain integrals of its positions. Take the integrand H_s = 2s + 5 on [0, T]:

Step 1 — split. Linearity peels the sum apart and pulls the constants out — exactly as for a Riemann integral, and for exactly the same reason: it holds term by term in the approximating sums \sum H_{t_i}\, \Delta W_i, and survives the limit.

Step 2 — recognise the easy leg. The constant integrand 1 just adds up the Brownian increments themselves: \int_0^T 1\, dW_s = W_T - W_0 = W_T. So

One opaque stochastic integral has become a known random variable plus a simpler integral we can analyse on its own (its variance is computed in Example 3 below). This decomposition — "write the exposure as a combination of legs you understand" — is the daily bread of every risk system: the Greeks of a book are the sum of the Greeks of its trades because the gain integral is linear.

Worked example 2 — zero mean: pure noise has no drift

A trader runs an adapted strategy \varphi_t — the position held at time t, decided using only information available at t — against a driftless price dS_t = \sigma\, dW_t. The terminal profit and loss is the gain integral

Zero — whatever the strategy, however ingenious, as long as it is adapted and square-integrable. Every dollar of expected gain in one scenario is exactly cancelled by expected loss in another, because each position \varphi_{t_i} is locked in before the mean-zero increment \Delta W_i it multiplies. An unhedged position in pure noise earns nothing on average: no adapted trading rule can conjure drift out of a driftless source.

Now bolt on a deterministic carry — a bank account, a funding leg — so the position is a drift plus noise:

Expectation surgically removes the stochastic-integral part and leaves the drift. This is the single most-used computation in the subject: whenever a process is written in the form "dX = (\text{drift})\, dt + (\text{noise})\, dW", its expected change is read off the dt term alone — the dW term contributes exactly nothing to the mean.

Worked example 3 — the variance: Var(∫₀ᵀ t dW) = T³/3

Zero mean does not make the position small — it makes its size a pure variance question, and the isometry answers it exactly. Compute the variance of the deterministic-ramp integral \int_0^T t\, dW_t (a position that grows linearly over time, like an accumulating hedge).

Step 1 — variance reduces to a second moment. The integral has mean zero, so no mean-squared term to subtract:

Step 2 — the isometry converts it to an ordinary integral. The Itô isometry swaps the mean-square of a stochastic integral for the plain time-integral of the squared integrand — here deterministic, so the outer expectation evaporates:

Two lines, no probability left in sight — the whole distributional question collapsed to first-year calculus. Compare it with the constant integrand: \operatorname{Var}(W_T) = \int_0^T 1\, dt = T. Ramping the position up as t loads the risk onto the late increments, and over a long horizon T^3/3 dwarfs T — a position that grows linearly accumulates risk cubically. (In fact \int_0^T t\, dW_t is Gaussian — a limit of sums of independent normals — so these two numbers, mean 0 and variance T^3/3, pin down its entire distribution: N(0, T^3/3).)

Continuity: a gain you can watch tick by tick

The last property on the list is easy to skate past: the path t \mapsto M_t is continuous with probability one. No jumps — ever. The integral inherits this from its driver: Brownian motion has continuous paths, and the running left-endpoint sums converge uniformly enough (that is the technical content) for the limit to keep the continuity. Trading a continuous price continuously produces a continuous gain.

Why care? Because continuity is what makes level-crossing arguments work. A continuous process cannot get past a barrier without touching it, so the first time M hits a level is a genuine stopping time at which M sits exactly on the level — the fact underlying barrier-option pricing, drawdown limits, and every stopped-martingale argument you will meet later. (Note what continuity does not give: the path is nowhere differentiable, just like the Brownian path underneath. Continuous, yes; smooth, never.)

One path of the integral — all the properties in one picture

Below is a sample path of M_t = \int_0^t H_s\, dW_s for a simple piecewise-constant deterministic integrand H, built as the running left-endpoint sum \sum_{t_i \le t} H_{t_i}\,\Delta W_i. Every property from the theorem is visible if you know where to look:

• Continuity — the path never jumps, however wildly it wiggles;
• Martingale / zero mean — it hovers around the dashed zero line, the fair-game level, never developing a drift no matter how far it wanders;
• The isometry, locally — H steps between values (from 0.8 up to 2), and the path is visibly calmer where H is small and jumpier where it is large: the local wiggle-energy is H_t^2\, dt, exactly the quadratic variation accumulating at rate H^2.

Refresh to draw a fresh path — the drawn path changes completely, but all three signatures persist, because they are properties of the construction, not of any one path.