advanced30 min studyLesson 673 in Mathematics

Itô formula and semimartingales

Derive the curvature correction and state the local integrability behind the differential shorthand.

Before you begin

Itô integral construction and isometry

On this page

The question for today

What you’ll understand

Apply the time-dependent Itô formula with its half-variance term.
Identify finite-variation and local-martingale components.
Check a transformed diffusion using local hypotheses rather than formal cancellation alone.

Ordinary Taylor expansion contains a second-order term that disappears along smooth paths. Brownian squared increments retain a time-sized limit, so that term survives as a drift correction. Itô's formula is the resulting chain rule. We work with continuous processes adapted to the usual filtration and with locally integrable drift and locally square-integrable Brownian coefficient. These local assumptions give meaningful stopped integrals even when a global expectation estimate is unavailable.

The semimartingale decomposition

A continuous semimartingale is a sum of a continuous local martingale and a continuous adapted finite-variation process, with an initial value specified. For an Itô process, the latter component is a time integral of the drift, while the local martingale is a Brownian stochastic integral. Assume the drift's absolute time integral and the volatility's squared time integral are finite almost surely on each finite horizon, or work up to stopping times where this is true.

The finite-variation component has zero continuous quadratic variation and zero covariation with the local-martingale component. Thus the diffusion's quadratic variation is the time integral of its squared volatility. The drift can strongly affect the path's distribution while contributing nothing to that bracket. Zero quadratic variation therefore does not mean a term is unimportant; it identifies how that term enters the chain rule and separates its first-order contribution from the noise's curvature correction.

X_t=X_0+\int_0^t b_s\,ds+\int_0^t\sigma_s\,dW_s,\qquad [X]_t=\int_0^t\sigma_s^2\,ds

The integrability assumptions are local in time and may be enforced by stopping.↔ Scroll wide equations horizontally.

Curvature creates an additional drift

For a deterministic function once continuously differentiable in time and twice in space, Itô's formula combines its time derivative, the ordinary drift derivative, and one half of its second spatial derivative times the local variance. The stochastic coefficient is its first spatial derivative times the original volatility. The formula holds as an identity of stopped stochastic integrals under the local assumptions. It does not itself assert that the stochastic term has expectation zero without the additional true-martingale integrability needed for that assertion.

For a general continuous semimartingale, the same formula uses its finite-variation differential, its local-martingale differential, and the bracket of the latter. Processes with jumps require extra jump corrections, which are excluded here. In several dimensions the curvature term contracts the Hessian with the covariance matrix. This covariance, rather than the number of independently written noise symbols, determines the second-order part of the generator.

df(t,X_t)=\left(f_t+b_tf_x+\tfrac12\sigma_t^2f_{xx}\right)(t,X_t)\,dt+\sigma_tf_x(t,X_t)\,dW_t

The extra half-variance term survives because the Brownian quadratic clock is nonzero.↔ Scroll wide equations horizontally.

Transformations must preserve their domain

The exponential transform provides a useful check: exponentiating Brownian motion with a compensating deterministic time term can cancel the Itô drift. Conversely, taking logarithms requires the process to remain positive, or requires stopping before it reaches zero. A formal logarithmic calculation across a boundary is invalid because the test function is not smooth there. Stopping inside compact subsets of its smooth domain makes the calculation legitimate before any boundary limit is considered.

Stratonovich notation obeys an ordinary-looking chain rule, but its coefficient conversion contains the correction that Itô displays explicitly. For a smooth scalar state-dependent volatility, the Itô drift equals the Stratonovich drift plus one half of volatility times its spatial derivative. The two notations describe the same process only after that conversion. Simply changing the integral symbol while retaining both coefficient functions changes the equation, except in special cases such as constant volatility.

Work it through

Square a Brownian motion with drift

Let $X_t=1+2t+3W_t$ and $Y_t=X_t^2$ . Derive the SDE for $Y$ , and calculate its expectation at time one. Check the expectation by expanding the original random variable directly.

For $f(x)=x^2$ , the first derivative is $2x$ and the second derivative is two. Substitute drift two and volatility three into Itô's formula.
The result is $dY_t=(4X_t+9)\,dt+6X_t\,dW_t$ . In particular, the number nine comes from one half times nine times two, not from squaring the drift.
Since $X$ has finite second moments on finite horizons, the displayed stochastic integral is a true square-integrable martingale. With $\mathbb EX_t=1+2t$ , integration gives $\mathbb EY_1=1+\int_0^1(4+8t+9)dt=18$ .
Directly, $X_1=3+3W_1$ , so its square has expectation $9+9=18$ . The direct calculation verifies both the curvature term and the initial value.

The result

The correct expected square is eighteen. Omitting the Itô correction would lose the contribution of the Brownian variance.

The living notebook

Explore the model

Interactive lab

Interactive lab · function

Inspect the curvature test function

Pause/Reset, then set both coefficient to 1 and position to 1. Predict value one and slope two for the quadratic test function, whose second derivative is two. Reset and set both coefficient to 0.5 and position to 1; the second derivative becomes one, halving its Itô curvature drift at fixed volatility.

y = ax²TangentSigned integral

y at chosen x: 0.608
Tangent slope · 2ax: 1.35
Signed area · ax³/3: 0.182

Coefficient a0.75Position x0.9

t = 0 s

This lab uses only y = ax². Playback traces the curve within the fixed integration interval from 0 to x; it does not estimate numerical error or change the chosen point. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThe quadratic panel exposes the derivatives used by Itô's formula. It neither draws a Brownian trajectory nor evaluates the random stochastic term.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Itô's formula gives d(W squared)=2W dW+dt, so subtracting time removes the drift. The stochastic integral is square integrable on finite horizons, which justifies the true-martingale conclusion.

Your turn

Try it yourself

For $Z_t=\exp(2W_t-2t)$ , derive its SDE and compute its expectation at every finite time. Explain why a local drift calculation alone would not generally suffice to establish the expectation.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Treat the exponent as a time-dependent test function of Brownian motion, and then use the Gaussian exponential moment separately.

Compare with the solution

The time derivative contributes $-2Z$ , while one half of the second Brownian derivative contributes $2Z$ . They cancel, leaving $dZ_t=2Z_t\,dW_t$ with initial value one. Gaussian integration gives $\mathbb EZ_t=e^{-2t}\mathbb E e^{2W_t}=e^{-2t}e^{2t}=1$ . The drift cancellation first gives a positive local martingale; the explicit integrability calculation is what verifies mean one and, here, the true-martingale property.

Keep the idea

Take it with you

Continuous finite variation contributes no quadratic clock.
Itô's second derivative term is multiplied by local variance.
A zero local drift does not eliminate the need for martingale integrability checks.

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