The first thing to be said about Itô’s Integral is that it isn’t really an integral in the same sense as an area under a curve basic calculus or Riemann rectangles integration way. An Itô Integral is really fundamentally a finance calculation.
It can and frequently does contain negative values. What sort of rectangle has a negative area?
So, set rectangles and areas aside. Instead, consider asset prices. The simplest asset to consider is a stock price. For simplicity, let’s begin our consideration of this stock’s price at $10 and allow this stock to increment or decrement in $1 intervals.
Over the course of a six day trading period, this stock’s price increased by $1 on Monday and again by another $1 on Tuesday. On Wednesday, it decreased by $1. On Thursday, it increased by $1. On Friday, it decreased by $1. The following Monday, it decreased by $1. This stock’s price fluctuations over this week were: +1, +1, -1, +1, -1, -1.
Imagine that we’re a stock trader, and we want to come up with a strategy for trading this stock. At time t, all we know is what’s happened at that time t, so whatever amount of stock we choose to buy (or sell) at time t cannot depend on information after time t. Let Wt be the total fluctuation in the stock up to time t. We get this by adding up the fluctuations up to that time. So W1=1, W2=2, W3=1, W4=2, W5=1, W6=0.
One simple trading strategy is to buy Wt shares at time t and sell them at time t+1. The idea here is that, if the stock has gone up a lot, then we should be more inclined to make a bet on it going up again, and if it’s gone down a lot, we should be inclined to bet on it going down again. [Note that Wt can be negative, which would correspond to selling the stock (short sales are allowed) and then buying it back again. If you’re not familiar with shorting stocks, just ignore this.]
The profit from this strategy is just the sum over the amount we buy at time t multiplied by fluctuation in price from time t to time t+1.
This looks like and is analogous to a Riemann sum, but it’s not, because the price can go down (have a negative value). So the base of the rectangle geometric conception of an integral makes no sense. And also, our Wt has to be strictly on the left-hand side of the interval from t to t+1, or else we’d be using future information.
Let’s look at the cumulative profit we would get if we did this strategy at each possible time point. After time 1, we bet W1=1 dollars, and the next fluctuation is +1, so our profit so far is 1×1=1. Next, we bet W2=2 dollars, but then the stock goes down, so our profit is 2×−1=−2. The overall profit so far is 1−2=−1. If we continue this, we get the following table:
| time t | 2 | 3 | 4 | 5 | 6 |
| cumulative profit by time t | 1 | -1 | 0 | -2 | -3 |
Now, compare these results to an integral of the function f(W) = Wt $$\frac{w_t^2}{2}$$:
| time t | 2 | 3 | 4 | 5 | 6 |
| cumulative profit by time t | 1 | -1 | 0 | -2 | -3 |
| Wt | 2 | 1 | 2 | 1 | 0 |
| \[\frac{w_t^2}{2}\] | 2 | 1/2 | 2 | 1/2 | 0 |
Notice what happens if you subtract t/2 from the integral. You get the cumulative profit:
| time t | 2 | 3 | 4 | 5 | 6 |
| cumulative profit by time t | 1 | -1 | 0 | -2 | -3 |
| Wt | 2 | 1 | 2 | 1 | 0 |
| \[\frac{w_t^2}{2}\] | 2 | 1/2 | 2 | 1/2 | 0 |
| \[\frac{w_t^2}{2}\, – \,\frac{t}{2}\] | 1 | -1 | 0 | -2 | -3 |
Therefore, for cumulative profit, we get the cumulative sum: \[\sum^{t} W_t(W_{t+1} \, – \, W_t) = \frac{w_t^2}{2} \, – \, \frac{t}{2}\] For any price change, this is the math underlying Itô calculus.
If you take an abstract limit, then time becomes continuous and Wt becomes Brownian motion. From there, the famous Itô formula of stochastic calculus emerges: $$\int_0^t \, W_t(dW) \, = \, \frac{W(t)^2}{2} \, -\frac{t}{2}$$
Both sides of this are random variables. The proof of this has nothing to do with the central limit theorem. It’s entirely because what we checked in the example above holds for any set of stock price fluctuations we could have chosen. This formula extends to functions of Wt. From these you can derive the chain rule of Itô calculus.
Thus, Itô’s integral is a way to calculate the random walk profit from trading a stock that moved randomly. In real life for financial professionals, Itô’s integral is a calculation method for incorporating a random process (Brownian motion) into a probabilistic decision. It is a more sophisticated technique than a simple weighted average of discrete variables that may or may not more accurately reflect and model underlying reality.
Wall Street certainly believes in stochastic calculus. In my experience, it is usually slightly more accurate than asking people for their uncertain beliefs about future outcomes, which is the common input to discrete variable weighted averages. Stochastic calculus, including Itô’s integral, is a vitally important precursor and input to what we now call data science.
In data science, we input all prior data like price fluctuations, as with stochastic calculus. However, in data science, it is only of secondary importance that we be able to model (explain) why movements/variations/changes occur. The primarily important goal is to predict their occurrence. When the random walk becomes less random, we want to predict its next step even though we cannot rationally explain why.