LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
Ito's lemma is the chain rule for a function of a random path. The ordinary chain rule drops every piece past the first derivative once the step is tiny. For Brownian motion, the square of the random step does not drop. It behaves like the time step, and that leftover is the Ito correction.
Why it matters
A derivative payoff is a function of an underlying price, and the usual model writes that price as a stochastic differential equation, a differential equation that includes noise. Passing from the underlying to the contract requires the differential of a function of that noise. The classroom derivative gets the average change wrong. The lecture's check uses more than 1,000 Brownian paths: their average change lines up with the Ito differential, while the Taylor differential that omits the correction sits far from that average.
The idea in plain words
In ordinary calculus the derivative is a limit, the slope of the tangent, and the tool behind differential equations. That limit stops being the right object once the input is random. A random variable is a distribution, with an expectation and a variance, not a single number.
A differential is a very small change, and it is not the number zero. Write it as the value after a bump minus the value before it. Send the bump toward zero and the differential shrinks. Substitute zero at the start and nothing has moved.
The Taylor series writes a smooth change as the function, plus the first derivative times the step, plus one half the second derivative times the step squared, and higher powers after that. Put that series into the limit definition and every leftover power of the step vanishes. The derivative is only the linear piece. A third-order truncation follows a bend for longer. The tangent peels away from a curving graph once you leave the contact point.
Time and Brownian motion change the bookkeeping. A small time step squared still vanishes. The product of a time step and a Brownian step still vanishes. The square of the Brownian step does not. It converges to the time step. That is the quadratic variation of Brownian motion.
The increment law is the reason. Over a short window the change in Brownian motion is normal with mean zero and variance equal to the window. Shrink the window and the variance is the infinitesimal time step. If the squared increment were zero, that variance would vanish with it. On a long interval from zero to a future time T, cut into pieces, the net change is normal with mean zero and variance T. If every squared increment were zero, the variance on that interval would be zero, which is not T. Add the time steps instead and the pieces sum to T, the variance the definition requires.
How to use it
Expand a function of time and of Brownian motion with the multivariable Taylor formula. Drop the squared time differential and the mixed product. Replace the squared Brownian differential with the time differential, and keep the one half that already sits on the second-derivative term. The collected result is the Ito differential: the time derivative times dt, plus the derivative with respect to the Brownian coordinate times the Brownian differential, plus one half the second derivative in that coordinate times dt.
The formula describes the average infinitesimal change, not the next wiggle on one path. The lecture compares, on a single path, the actual small change, the Ito differential, and the ordinary Taylor differential. They do not coincide. One path is one draw. Average the actual change across more than 1,000 paths and that average sits on the Ito curve. The version that sends the squared Brownian piece to zero misses the average. The law of large numbers is why more paths tighten the match: the sample mean of the increments settles on the mean the lemma names.
What usually goes wrong
Deleting every second-order term is correct for a smooth ordinary function and wrong for Brownian motion, because that deletion throws away the variance. The gap on one simulated path is intentional. The matching object is the mean differential. The correction is the Taylor term you are no longer allowed to drop, and the same expansion later turns a stochastic differential equation into a partial differential equation for a contract.
Questions people ask
Why does the Brownian square survive when the time square does not?
A Brownian increment over a short window has variance equal to that window, so its square carries variance. A pure time increment has no spread, and its square vanishes in the limit.
Does the lemma predict the next increment?
It characterizes the mean of the change. One simulated path is not supposed to match the formula exactly.
What did the path average show?
Across more than 1,000 draws, the average realized differential lined up with Ito's formula. The Taylor differential without the squared-Brownian correction did not.