Itô's integral

Intuitive idea

Consider a game that consists of tossing a coin, and you win if it comes up heads and lose if it comes up tails, in a succession of tosses. Let's call $X_{t}$ to the result of the toss (codified as 1 or -1) at time $t \in {1, 2, 3, \dots}$ . The expression

P_{t} = {\begin{cases} 0, if t = 0 \\ \sum_{i = 1}^{t} X_{i}, if t > 0 \end{cases}

is a stochastic process called a random walk.
Now suppose you have a simple betting strategy which consists of betting an amount $H_{i}$ at the tosses $i + 1$ , for example $H_{i} = 2 (i + 1)$ . You compute your profit with the expression

B_{t} = \sum_{i = 0}^{t} H_{i} \cdot (P_{i + 1} - P_{i})

Now, if we rescale time (we have a coin toss every instant of time), the random walk converges to Brownian motion. The profit sums then become

\sum H (s_{i}) (W_{t_{i + 1}} - W_{t_{i}}),

with $s_{i} \in [t_{i}, t_{i + 1}]$ . The value of $H (s_{i})$ is generalized to depend, not only on $i$ but on the outcomes up to time $s_{i}$ .
This looks like Riemann–Stieltjes sums but with random increments. Taking the limit as the mesh goes to zero defines the Itô integral

\int_{0}^{t} H (s) d W_{s} .

Ideas for later:

In Itô’s definition, we always take $s_{i} = t_{i}$ (the left endpoint).

In the Stratonovich integral, you instead take $s_{i} = \frac{t_{i} + t_{i + 1}}{2}$ (the midpoint).
In ordinary Riemann--Stieltjes integral you could in principle choose any point in $[t_{i}, t_{i + 1}]$ , and it wouldn’t matter if the integrator were smooth. But for Brownian motion it does matter, because the path is too irregular.

Two more examples in discrete setup:

Stock portfolio

$W_{t}$ : random "shocks" of the market.
$H (t)$ : number of shares you hold at time $t$ .
$Δ W_{i}$ : unpredictable change in the market in each interval.
Then

X_{t} = \sum_{i} H (t_{i}) Δ W_{i}

is the random gain/loss of your portfolio.

At the end of the day ( $t$ fixed), you don’t get a single predictable value — you get a distribution of possible outcomes. Over time, $(X_{t})_{t \geq 0}$ is the whole stochastic process of your wealth.

Boat on a choppy sea

$W_{t}$ : waves hitting the boat (random but with variance growing with time).
$H (t)$ : how much sail you expose at time $t$ .
The Itô integral is the random displacement of your boat due to all the wave kicks, scaled by your sail choice.

If $H (t)$ is large, you’re more sensitive to randomness. If $H (t) = 0$ , you ignore the waves. See a simulation in https://ajpancollantes.github.io/cajondes/sdeboat.html.

In general

Suppose time is chopped into little steps $Δ t$ . Brownian motion increments look like

Δ W_{i} \sim N (0, Δ t),

independent for each step.

If you have a rule $H (s)$ that tells how strongly you respond to randomness at time $s$ , then the Itô integral is approximated by

X_{t} = \int_{0}^{t} H (s) d W_{s} \approx \sum_{i} H (t_{i}) Δ W_{i} .

Key takeaway

The Itô integral is:

Not a number, but a random variable for each time $t$ .
Over time, it builds a new stochastic process.
Discretely, it’s like a weighted random walk: sum of (weight × random increment).

About the integrand $H$

In the Itô integral setup we have three “players”:

$W_{t}$ – Brownian motion (the noise, external randomness).
- This is a stochastic process with independent increments, variance proportional to time, etc.
$H_{t}$ – the integrand / strategy process.
- This is also a stochastic process in general.
- Importantly: $H_{t}$ must be adapted to the filtration of $W_{t}$ .
  
  Meaning: at time $t$ , your value of $H_{t}$ can depend on all information from the past and present ${W_{s} : s \leq t}$ , but not on the future.
- So $H_{t}$ could be deterministic (e.g. $H_{t} \equiv 2$ ), or it could respond to past values of $W$ (e.g. “double the sail if the last wave was positive”), which makes it a stochastic process.
$X_{t} = \int_{0}^{t} H (s) d W_{s}$ – the resulting Itô integral process.
- This is the new stochastic process built from combining $H$ and $W$ .
- For each fixed $t$ , $X_{t}$ is a random variable.
- As $t$ varies, $(X_{t})_{t \geq 0}$ is a stochastic process.
- Its distribution depends on both the randomness of $W$ and the way $H$ responds to it.

In practice, $H_{t}$ can depend on any information available up to time $t$ . This is called being adapted to the filtration of the driving process (usually $W_{t}$ or some stochastic process representing the noise). Let’s break this down in practical terms.

$H_{t}$ can be a fixed function of time only: $H_{t} = f (t)$ .
- Example: for the boat, “keep sail at half size all the time,” or for a stock, “always hold 10 shares.”
- No dependence on randomness — simplest case.
- $H_{t} = 1$ recovers plain Brownian motion: $\int_{0}^{t} d W_{s} = W_{t}$ .
$H_{t}$ can use the past trajectory of $W$ : $H_{t} = g (W_{s}, s \leq t)$ .
- Examples in our analogies:
  - Boat: adjust sail according to previous wave heights or cumulative displacement.
    - “If last wave was +1, increase sail; if -1, reduce sail.”
  - Stock: number of shares held depends on past stock price or returns.
    - “If yesterday the stock went up, sell; if down, buy.”
- Key: $H_{t}$ cannot depend on future increments ( $W_{t + Δ}$ ), only on what’s observable up to $t$ .
If there are other stochastic processes $Y_{t}$ that you can observe, can also depend on them, as long as they’re measurable at time $t$ .
- Example:
  * Weather: boat adjusts sail depending on past wind speed (another stochastic process).
  * Finance: option hedging might depend on both past stock price $S_{t}$ and interest rate $r_{t}$ .
Often, $H_{t}$ depends on a summary statistic of the past:
- Cumulative sum: $S_{t} = \int_{0}^{t} H_{s} d W_{s}$ (current wealth or displacement).
- Running maximum or minimum: e.g., for stop-loss strategies.
- Moving averages: smooth history of past observations.
- These are all valid as long as they only use past information.

Next step: SDEs

Ito's integral

X_{t} = \int_{0}^{t} H (s) d W_{s}

corresponds to the simplest case of stochastic ordinary differential equation:

d X_{t} = H_{t} d W_{t} .

So now we can consider that we introduce a deterministic component, known as the drift:

d X_{t} = a (t, X_{t}) d t + H (t, X_{t}) d W_{t}

Think of the change in a system, $d X_{t}$ , over a tiny time step as being composed of two parts:

A predictable, deterministic part (the "drift"): This is the $a (t, X_{t}) d t$ term.
An unpredictable, random part (the "diffusion"): This is the $b (t, X_{t}) d W_{t}$ term.

In integral form, this SDE is equivalent to:

X_{t} = X_{0} + \int_{0}^{t} a (s, X_{s}) d s + \int_{0}^{t} b (s, X_{s}) d W_{s}

Here, the first integral is a standard Riemann integral (the predictable path), and the second is the Itô integral above (the random path).

An Intuitive Example: A Drifting Boat
Let's revisit the boat on a choppy sea analogy. With the new SDE model: $d X_{t} = a (t, X_{t}) d t + b (t, X_{t}) d W_{t}$ we have a more realistic boat:
* The drift term $a (t, X_{t}) d t$ could represent the boat's engine and the current. If you point the boat east and turn on the engine, you introduce a predictable eastward drift. $a (t, X_{t})$ is the speed and direction of this deterministic movement.
* The diffusion term $b (t, X_{t}) d W_{t}$ still represents the random kicks from the waves. $b (t, X_{t})$ is your sail setting, controlling how much the waves affect you.

So, the total movement of the boat is the sum of where its engine is taking it (predictable drift) and how the waves are knocking it about (random diffusion). See the simulation at https://ajpancollantes.github.io/cajondes/sdeboat.html.