LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
An ordinary integral returns one number, the signed area under a curve. An Ito integral returns a random variable. It is the running sum of Brownian increments, each multiplied by a weight, so a new draw of the noise produces a new value of the integral.
Why it matters
Desk models are stochastic differential equations. A closed form can be reused in a later formula and calibrated without resimulating every trial. Many integrals have none, so you discretize. Either route has to draw from the same distribution. Pricing also cares whether the integral is a martingale, which is where Ito and Stratonovich integration part ways.
The idea in plain words
The signed area under y = x squared from 0 to 2 is 8/3, about 2.667. Very wide rectangles in the lecture gave an area near 1. Shrink the width and they approach 8/3. A derivative step of 0.00001 is already close for the same reason.
A random variable is not a stand-in for one number. A standard normal is a density of outcomes. Square it and the square is a random variable too, because you cannot know x squared while x itself is unsettled. An integral against Brownian motion inherits that fact. You can estimate its mean or its variance. You cannot replace the integral with one deterministic area.
Brownian motion starts at zero, is continuous with probability one, has independent increments, and those increments are normal with variance equal to the step. Add the increments and you get the path. Anything built from that path, including its differential and its integral, is still a draw from a distribution.
Ito's lemma splits the change in a function of Brownian motion into a drift piece and a shock piece. The extra piece, relative to classroom calculus, is there because the squared Brownian increment behaves like elapsed time. On the lecture's plots, the actual small change in W squared and the lemma's decomposition sit together on each path, up to the error of the time step. A new path changes both curves together.
How to use it
Read the integral as a weighted cumulative sum. Brownian increments are the noise. The integrand is the weight. In the lecture's picture the weight is time, drawn as a surface. Lift the path onto that surface and sum the weighted increments. The resulting curve is one realized value. Run it again and the curve changes. Average many paths and the law of large numbers supplies a numerical mean and variance, if the discretization is the integral you intended.
A closed form, when you can find one, still has to contain noise. The lecture's recipe is to guess a candidate, apply Ito's lemma, rearrange, and integrate. For the integral of W with respect to W, the candidate is W squared. The time derivative is zero. The first derivative in W is 2W. The second is 2. Substitute, integrate both sides, and divide by two. The integral equals one half of W squared minus time. The starting value of Brownian motion is zero, so the constant from the lower limit drops out. Randomness remains on the right-hand side, which is required: a deterministic answer would describe a different object.
Discretize the formula and the original integral with an Euler–Maruyama step. As the time step shrinks, the lecture's curves overlap and the gap falls. The closed form can enter a later equation. The numerical path cannot. When no formula exists, that path is still a valid sample, and often the only one.
What usually goes wrong
The first mistake is to expect the integral to print one number the way 8/3 does. A single simulated path is one draw. The second is to drop the Ito correction because a midpoint, Stratonovich, construction does not show it. Ito integration uses the left edge of each rectangle, so the weight is known before the increment arrives. That makes the integral adapted to the information so far, and a martingale. The midpoint looks ahead, so the Stratonovich integral is not a martingale. Risk-neutral pricing is built on that martingale property. Using the prettier calculus and then treating the result as a hedgeable price mixes two different objects.
Questions people ask
Why is the answer one half of W squared minus time?
That is what remains after Ito's lemma is applied to W squared and both sides are integrated. The minus-time term is the quadratic-variation correction. Without it the variance is wrong.
Do the formula and the simulation agree?
In the lecture they approach each other as the time step gets small. Both are samples from the integral, so they match in distribution rather than tick for tick on every coarse grid.
When is the closed form worth the work?
When you can drop it into another equation, or when calibration has to be fast. If the contract needs the whole path and no formula exists, discretize the integral directly.