Chapter 05

Rigorous

The Itô Integral

00 · Symbol Glossary

$W_t$W sub t — standard Brownian motion

The driving noise from Chapter 02: $W_0=0$ , independent increments, $W_t-W_s\sim N(0,t-s)$ for $s<t$ , continuous paths. Everything in this chapter integrates against $W_t$ .

$H_t$integrand — an adapted process

The process being integrated. $H_t$ must be adapted to the filtration $\mathcal{F}_t$ of Chapter 04 — it cannot peek at the future value of $W$ .

$\int_0^T H_t\,dW_t$Itô integral

A random variable built as a limit of sums $\sum_i H_{t_i}(W_{t_{i+1}}-W_{t_i})$ , evaluated at the left endpoint of each subinterval. The left-endpoint choice is not cosmetic — it is what makes the whole theory work.

$L^2$square-integrable processes

The space of processes with $\mathbb{E}\!\int_0^T H_t^2\,dt<\infty$ . The Itô integral is first built for simple processes, then extended to all of $L^2$ by a limiting argument.

01 · Why an Ordinary Integral Does Not Work

Chapter 03 showed that $W_t$ has infinite total variation on every interval, no matter how short. A Riemann–Stieltjes integral $\int_0^T H_t\,dW_t$ is normally defined path by path, as a limit of sums that converges regardless of which point in each subinterval you sample — left endpoint, right endpoint, midpoint, all give the same answer when the integrator has finite variation. Brownian motion does not have finite variation, so that convergence fails: different sampling points give different limits, and none of them exist as an ordinary limit for almost every path.

The fix is not to abandon the sum. It is to fix the sampling point once and for all — always the left endpoint — and build convergence in a weaker sense than "for almost every path": convergence in $L^2$ , i.e. in mean square across the randomness. That single choice is the entire content of Itô's construction.

Why left, specifically

The left endpoint makes $H_{t_i}$ known at the start of $[t_i,t_{i+1}]$ — no information about the future increment $W_{t_{i+1}}-W_{t_i}$ leaks into the weight. That is exactly the "adapted, non-anticipating" requirement, and it is what produces the martingale and isometry properties in Section 03. Change the sampling point and both properties disappear.

02 · Step One: Simple (Piecewise-Constant) Processes

Build the integral first for the easiest integrands, where no limit is needed at all.

Definition — Simple Process

A process $H_t$ is simple if there is a partition $0=t_0<t_1<\cdots<t_n=T$ and $\mathcal{F}_{t_i}$ -measurable random variables $H_i$ (each with finite variance) such that

H_t = H_i \quad \text{for } t\in[t_i,t_{i+1})

$H_t$ is constant on each subinterval, and that constant is known already at the start of the subinterval.

For a simple process, define the integral directly as the sum that motivated it:

\int_0^T H_t\,dW_t \;:=\; \sum_{i=0}^{n-1} H_i\,(W_{t_{i+1}}-W_{t_i})

There is no limit here — it is a finite sum of random variables, each term a known weight times a Brownian increment. This object already has the two properties that will survive the general construction, and they are worth checking directly. $H_i$ is measurable with respect to $\mathcal{F}_{t_i}$ , and $W_{t_{i+1}}-W_{t_i}$ is independent of $\mathcal{F}_{t_i}$ with mean zero, so $\mathbb{E}[H_i(W_{t_{i+1}}-W_{t_i})\mid\mathcal{F}_{t_i}]=H_i\cdot\mathbb{E}[W_{t_{i+1}}-W_{t_i}\mid\mathcal{F}_{t_i}]=0$ . Sum that over $i$ and the whole integral has conditional mean zero at every stage — the martingale property, checked from the definition rather than assumed.

Example — A two-step simple integral

Partition $[0,2]$ at $t_0=0,t_1=1,t_2=2$ , with $H_t=W_0=0$ on $[0,1)$ and $H_t=W_1$ on $[1,2)$ . Then

\int_0^2 H_t\,dW_t = 0\cdot(W_1-W_0) + W_1\cdot(W_2-W_1) = W_1(W_2-W_1)

$\mathbb{E}[W_1(W_2-W_1)]=\mathbb{E}[W_1]\,\mathbb{E}[W_2-W_1]=0$ by independence of the increment from $W_1$ — consistent with the martingale property above.

03 · Extension to $L^2$

Most processes worth integrating — $H_t=W_t$ , $H_t=f(W_t,t)$ — are not piecewise constant. The extension is a standard density argument: approximate, take a limit, check the limit does not depend on the approximating sequence.

Step-by-step — Building the general Itô integral

Restrict to $L^2$ : require

\mathbb{E}\int_0^T H_t^2\,dt<\infty

and

H_t

adapted. This is the class the construction covers; wider classes exist but need more machinery.

Approximate: for any such

H

, there is a sequence of simple processes

H^{(n)}

with

\mathbb{E}\int_0^T (H_t^{(n)}-H_t)^2\,dt\to 0

— simple processes are dense in

L^2

of adapted processes.

Define via the limit:

\int_0^T H_t^{(n)}\,dW_t

is already defined (Section 02) and forms a Cauchy sequence in

L^2(\Omega)

because of the isometry below. Define

\int_0^T H_t\,dW_t

as its

L^2

limit.

Check well-posedness: any two approximating sequences give the same limit, so the integral is a well-defined random variable, not an artifact of the approximation chosen.

The isometry that Step 3 leans on is the single most useful computational fact in the whole subject.

Definition — Itô Isometry, Martingale Property, Zero Mean

For $H\in L^2$ adapted,

\mathbb{E}\!\left[\left(\int_0^T H_t\,dW_t\right)^2\right] = \mathbb{E}\!\int_0^T H_t^2\,dt \qquad \textbf{(Itô isometry)}

\mathbb{E}\!\left[\int_0^T H_t\,dW_t\right] = 0 \qquad \textbf{(zero mean)}

and $M_t=\int_0^t H_s\,dW_s$ is a martingale: $\mathbb{E}[M_t\mid\mathcal{F}_s]=M_s$ for $s\leq t$ .

Plain language: the isometry converts a variance computation on a stochastic integral into an ordinary (deterministic-looking) integral of $\mathbb{E}[H_t^2]$ — no need to expand a double sum of Brownian increments by hand. The martingale property says the running integral $M_t$ has no drift: knowing everything up to time $s$ , the best forecast of $M_t$ is just $M_s$ . Both facts trace back to the left-endpoint, no-look-ahead sampling of Section 02 — they are inherited, not new assumptions.

Example — Variance of $\int_0^T W_t\,dW_t$

By the isometry, $\mathrm{Var}\!\left(\int_0^T W_t\,dW_t\right)=\mathbb{E}\int_0^T W_t^2\,dt=\int_0^T \mathbb{E}[W_t^2]\,dt=\int_0^T t\,dt=T^2/2$ . Chapter 06 will show $\int_0^T W_t\,dW_t=\tfrac12 W_T^2-\tfrac12 T$ exactly, and it is a good check to confirm $\mathrm{Var}(\tfrac12 W_T^2-\tfrac12 T)=\tfrac12 T^2$ matches — a preview of Itô's Lemma.

❌ Treating $\int_0^T W_t\,dW_t$ as ordinary calculus

Naive substitution suggests $\int_0^T W_t\,dW_t=\tfrac12 W_T^2$ , by analogy with $\int_0^T x\,dx=\tfrac12 x^2$ .

Why it breaks: ordinary calculus assumes the integrator has finite variation and that the sampling point in the defining sum does not matter (Section 01). Both fail for $W_t$ . The correction term comes from the nonzero quadratic variation $[W]_T=T$ that Chapter 03 established.

Consequence: the correct identity is $\int_0^T W_t\,dW_t=\tfrac12 W_T^2-\tfrac12 T$ . Dropping the $-\tfrac12 T$ misstates the mean (the naive guess has $\mathbb{E}[\tfrac12 W_T^2]=\tfrac12 T\neq 0$ , contradicting the zero-mean property above) and every downstream Greek or hedge computed from it.

04 · Itô vs. Stratonovich

Two integrals, two conventions

The Stratonovich integral samples at the interval midpoint instead of the left endpoint, and it obeys ordinary chain-rule calculus — no correction terms. It is used in physics, where noise is often a limit of smooth processes and the midpoint arises naturally from that limit. Finance almost always uses Itô: the left-endpoint convention matches "you trade at today's price using only information available today," which is exactly the non-anticipating requirement of Section 01. The two integrals differ by a deterministic correction term and are related by a conversion formula, but mixing them in one derivation — using an Itô SDE with Stratonovich calculus rules, or vice versa — silently reintroduces the finite-variation assumption that Section 01 showed fails for $W_t$ .

05 · Exercises

EXERCISE 5.1

Apply the isometry directly with $H_t=t$ .

$\mathrm{Var}\!\left(\int_0^T t\,dW_t\right)=\mathbb{E}\int_0^T t^2\,dt=\int_0^T t^2\,dt=T^3/3$ . The mean is $0$ by the zero-mean property, since $H_t=t$ is deterministic (hence trivially adapted) and square-integrable.

Compute $\mathrm{Var}\!\left(\int_0^T t\,dW_t\right)$ using the Itô isometry.

EXERCISE 5.2

Check the conditional-mean-zero argument of Section 02 for a two-step simple process, but change which endpoint of the subinterval is sampled.

Right-endpoint sampling would use $H_i'=H_{t_{i+1}}$ , which is $\mathcal{F}_{t_{i+1}}$ -measurable, not $\mathcal{F}_{t_i}$ -measurable. Then $\mathbb{E}[H_{t_{i+1}}(W_{t_{i+1}}-W_{t_i})\mid\mathcal{F}_{t_i}]$ no longer factors into a product of a known constant and a zero-mean increment, because $H_{t_{i+1}}$ and the increment can be correlated (e.g. $H_t=W_t$ gives $\mathbb{E}[W_{t_{i+1}}(W_{t_{i+1}}-W_{t_i})]=t_{i+1}-t_i\neq 0$ ). The martingale property breaks.

Explain why replacing the left endpoint with the right endpoint in the defining sum of Section 02 destroys the martingale property, using the conditional-expectation argument from that section.

EXERCISE 5.3

Compare the two conventions' treatment of $\int_0^T W_t\,dW_t$ ; Stratonovich gives the ordinary-calculus answer.

Itô: $\int_0^T W_t\,dW_t=\tfrac12 W_T^2-\tfrac12 T$ (Section 03). Stratonovich: $\int_0^T W_t\circ dW_t=\tfrac12 W_T^2$ , matching ordinary calculus, since the midpoint sampling absorbs exactly the quadratic-variation correction that left-endpoint sampling leaves exposed. The difference is $\tfrac12 T$ — a deterministic quantity, not noise — which is the general pattern relating the two integrals.

State the Stratonovich value of $\int_0^T W_t\circ dW_t$ and compare it to the Itô value from Section 03. What kind of quantity is the difference?

06 · Chapter Summary

Concept	Meaning
Simple process	piecewise-constant, known at the start of each subinterval
Left-endpoint sampling	encodes "no look-ahead"; source of the martingale property
$L^2$ extension	limit of simple-process integrals in mean square
Itô isometry	$\mathbb{E}[(\int H\,dW)^2]=\mathbb{E}\int H^2\,dt$
Zero mean / martingale	$\int_0^t H_s\,dW_s$ has no drift
Itô vs. Stratonovich	left vs. midpoint sampling; finance uses Itô

Next: Chapter 06 — Itô's Lemma, the chain rule that accounts for the $-\tfrac12 T$ correction term seen in $\int_0^T W_t\,dW_t$ and generalizes it to any smooth function of $W_t$ .