Chapter 2 — Construction of the Itô Integral

Guiding question

Brownian motion is nowhere differentiable. Its total variation is almost surely infinite. Classical Riemann–Stieltjes integration requires the integrator to have bounded variation. How, then, can we define $\int_0^T H_t\,dW_t$ in a way that is mathematically precise and financially meaningful?

The Itô integral is the central technical object of stochastic calculus. Chapter 1 built Brownian motion and established its quadratic variation property $[W]_T = T$. This chapter uses that property as a foundation. The strategy is:

1. Observe that pathwise Riemann–Stieltjes integration fails.
2. Define the integral for a small, explicitly computable class (simple processes).
3. Prove an isometry that makes the integral a bounded linear map in $L^2$.
4. Extend to the full $L^2$ class by density and closure.

The construction mirrors the standard $L^2$ completion in functional analysis, and the isometry is its counterpart of Parseval's theorem.

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2.1 Why pathwise integration fails

Let $f:[0,T]\to\mathbb R$ be a continuous function and $g:[0,T]\to\mathbb R$ be the integrator. The Riemann–Stieltjes integral $\int_0^T f\,dg$ exists whenever $f$ is continuous and $g$ has bounded variation:

$$\|g\|_{\mathrm{TV}} = \sup_\Pi \sum_{k}|g(t_{k+1})-g(t_k)| < \infty.$$

For Brownian motion, the opposite holds.

Brownian motion has infinite total variation

Almost surely,

$$\sum_{k=0}^{n-1}|W_{t_{k+1}}-W_{t_k}|\to\infty$$

as the mesh $|\Pi|\to 0$.

Proof

On a uniform partition with $t_k = kT/n$, the increments are i.i.d.\ $N(0,T/n)$. Therefore

$$\mathbb E\left[\sum_{k=0}^{n-1}|W_{t_{k+1}}-W_{t_k}|\right] = n\cdot \mathbb E\left[|N(0,T/n)|\right] = n\cdot \sqrt{\frac{2T}{\pi n}} = \sqrt{\frac{2nT}{\pi}} \to\infty.$$

By the law of large numbers applied to the i.i.d.\ terms $|W_{t_{k+1}}-W_{t_k}|$, the sum converges almost surely to $\infty$.

So Brownian paths are nowhere near having bounded variation. No classical integral theory applies directly.

Why infinite variation but finite quadratic variation?

Total variation sums $|\Delta W|$. Quadratic variation sums $(\Delta W)^2$. Because $\Delta W \sim \sqrt{\Delta t}$, we have

$$|\Delta W| \sim \sqrt{\Delta t}, \quad (\Delta W)^2 \sim \Delta t.$$

Summing over $n = T/\Delta t$ terms:

$$\sum |\Delta W| \sim n\sqrt{\Delta t} = \sqrt{T/\Delta t}\to\infty, \quad \sum (\Delta W)^2 \sim n\cdot\Delta t = T.$$

Quadratic variation lives in the right scaling window to remain finite. This is why $(dW)^2 = dt$ is the key equation, and why the natural Hilbert space for stochastic integration is built on the $L^2$ norm.

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2.2 The class of simple adapted processes

A process $H = (H_t)_{0\le t\le T}$ is called simple (or elementary) if there exists a partition $0 = t_0 < t_1 < \cdots < t_n = T$ and $\mathcal F_{t_k}$-measurable random variables $H_k$ satisfying $\mathbb E[H_k^2]<\infty$, such that

$$H_t = \sum_{k=0}^{n-1} H_k\,\mathbf{1}_{(t_k,t_{k+1}]}(t).$$

The requirement that $H_k$ is $\mathcal F_{t_k}$-measurable is the adaptedness condition: the value of the integrand at time $t_k$ may only use information up to $t_k$.

Adaptedness is not optional

If $H_k$ were allowed to depend on $W_{t_{k+1}}$ — the future increment — the resulting integral would not be a martingale, and the isometry below would fail. In finance, this corresponds to using future prices to choose a present position: pure look-ahead bias.

For a simple process, we define the Itô integral as the finite sum

$$I(H) = \int_0^T H_t\,dW_t := \sum_{k=0}^{n-1} H_k\bigl(W_{t_{k+1}}-W_{t_k}\bigr).$$

Each term is a product of a $\mathcal F_{t_k}$-measurable random variable with an independent increment. This is a finite sum of square-integrable random variables, so $I(H)\in L^2(\Omega)$.

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2.3 The Itô isometry for simple processes

Itô isometry

For any simple adapted process $H$,

$$\mathbb E\!\left[\!\left(\int_0^T H_t\,dW_t\right)^{\!2}\right] = \mathbb E\!\left[\int_0^T H_t^2\,dt\right].$$

Proof

Write $\Delta W_k = W_{t_{k+1}}-W_{t_k}$ and $\Delta t_k = t_{k+1}-t_k$. Then

$$\left(\sum_{k=0}^{n-1} H_k\,\Delta W_k\right)^{\!2} = \sum_{k=0}^{n-1} H_k^2\,(\Delta W_k)^2 + 2\sum_{j<k} H_j\,\Delta W_j\,H_k\,\Delta W_k.$$

Cross terms. For $j < k$, the product $H_j\,\Delta W_j\,H_k$ is $\mathcal F_{t_k}$-measurable, and $\Delta W_k$ is independent of $\mathcal F_{t_k}$ with mean zero. Therefore

$$\mathbb E[H_j\,\Delta W_j\,H_k\,\Delta W_k] = \mathbb E[H_j\,\Delta W_j\,H_k]\cdot\mathbb E[\Delta W_k] = 0.$$

Diagonal terms. For each $k$, $H_k$ is $\mathcal F_{t_k}$-measurable and independent of $\Delta W_k$. Therefore

$$\mathbb E[H_k^2\,(\Delta W_k)^2] = \mathbb E[H_k^2]\cdot\mathbb E[(\Delta W_k)^2] = \mathbb E[H_k^2]\cdot\Delta t_k.$$

Summing,

$$\mathbb E\!\left[\left(\int_0^T H_t\,dW_t\right)^{\!2}\right] = \sum_{k=0}^{n-1}\mathbb E[H_k^2]\,\Delta t_k = \mathbb E\!\left[\int_0^T H_t^2\,dt\right].$$

The cross terms vanish because of adaptedness. This is the structural insight: the left endpoint rule is exactly what kills the cross terms.

Isometry: the integral is an isometric embedding

The map $H\mapsto I(H)$ sends the Hilbert space

$$L^2([0,T]\times\Omega, dt\otimes\mathbb P)$$

isometrically into $L^2(\Omega,\mathbb P)$. Distances are preserved:

$$\|I(H)\|_{L^2(\Omega)} = \|H\|_{L^2([0,T]\times\Omega)}.$$

This is the stochastic counterpart of the identity $\|\hat f\|_2 = \|f\|_2$ in Fourier analysis: an integral transform that preserves $L^2$ norms.

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2.4 Zero mean

Itô integral is centered

For any simple adapted $H$,

$$\mathbb E\!\left[\int_0^T H_t\,dW_t\right] = 0.$$

Proof

Each term: $\mathbb E[H_k\,\Delta W_k] = \mathbb E[H_k]\cdot\mathbb E[\Delta W_k] = 0$, since $H_k$ is independent of $\Delta W_k$ and $\mathbb E[\Delta W_k]=0$.

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2.5 Extension to $L^2$ adapted processes

Let $\mathcal L^2 = L^2_{\mathrm{ad}}([0,T]\times\Omega)$ denote the space of adapted processes $H$ with

$$\|H\|^2 = \mathbb E\!\left[\int_0^T H_t^2\,dt\right] < \infty.$$

Density of simple processes

The simple adapted processes are dense in $\mathcal L^2$: for any $H\in\mathcal L^2$, there exists a sequence of simple processes $H^{(n)}$ with

$$\|H^{(n)}-H\|_{\mathcal L^2}\to 0.$$

Given this density, define the Itô integral for general $H\in\mathcal L^2$ as the $L^2(\Omega)$-limit:

$$\int_0^T H_t\,dW_t = \lim_{n\to\infty}\int_0^T H_t^{(n)}\,dW_t,$$

where the limit is in $L^2(\Omega)$.

Well-definedness

The sequence $I(H^{(n)})$ is Cauchy in $L^2(\Omega)$ because, by the isometry,

$$\|I(H^{(m)})-I(H^{(n)})\|_{L^2(\Omega)} = \|I(H^{(m)}-H^{(n)})\|_{L^2(\Omega)} = \|H^{(m)}-H^{(n)}\|_{\mathcal L^2}\to 0.$$

Since $L^2(\Omega)$ is complete, the limit exists and is independent of the approximating sequence.

Itô integral as a left-endpoint Riemann sum over Brownian motion

Partition size n = 40

Itô sum (left endpoint)0.8162

Stratonovich sum (midpoint)1.2593

Itô formula: ½(W₁² − 1)0.7593

Blue rectangles use H(t_k) — left endpoint, Itô convention. The Itô sum converges to ½(W₁² − 1), not ½W₁², because (dW)² = dt contributes an extra −½ correction — the Itô correction term. The Stratonovich midpoint sum converges to ½W₁².

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2.6 The Itô integral as a martingale

Define the running integral

$$M_t = \int_0^t H_s\,dW_s, \qquad 0\le t\le T.$$

Martingale property

The process $(M_t)_{0\le t\le T}$ is a martingale with respect to the filtration $(\mathcal F_t)$.

Proof sketch for simple processes

For $s < t$ and a simple process adapted to a partition that includes $s$,

$$M_t - M_s = \sum_{t_k\in[s,t)} H_k\,\Delta W_k.$$

Each term $H_k\,\Delta W_k$ has mean zero conditionally on $\mathcal F_s$ (since $H_k$ is $\mathcal F_{t_k}\supseteq\mathcal F_s$-measurable and $\Delta W_k$ is independent of $\mathcal F_{t_k}$ with mean zero). Therefore

$$\mathbb E[M_t - M_s\mid\mathcal F_s] = 0, \qquad\text{so}\quad \mathbb E[M_t\mid\mathcal F_s] = M_s.$$

Trading P&L is a martingale under no-arbitrage

If $H_t$ is the number of shares of a zero-drift asset held at time $t$, and the asset price satisfies $dS_t = \sigma_t\,dW_t$ (a pure diffusion under the risk-neutral measure), then the portfolio value

$$P_t = P_0 + \int_0^t H_s\,dS_s = P_0 + \int_0^t H_s\sigma_s\,dW_s$$

is a martingale. The expected P&L is zero: there is no predictable profit without risk. This is the mathematical content of no-arbitrage in diffusion models.

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2.7 Quadratic variation of the Itô integral

Quadratic variation

Let $M_t = \int_0^t H_s\,dW_s$. Then the quadratic variation of $M$ is

$$[M]_t = \int_0^t H_s^2\,ds.$$

Proof sketch

On a uniform partition with mesh $\Delta t$,

$$\sum_{k}(M_{t_{k+1}}-M_{t_k})^2 \approx \sum_k H_{t_k}^2\,(\Delta W_k)^2 \approx \sum_k H_{t_k}^2\,\Delta t \to\int_0^t H_s^2\,ds.$$

The approximation uses $(dW)^2 = dt$ in the same sense as the quadratic variation of Brownian motion.

Volatility as accumulated integrand squared

The quadratic variation $[M]_t = \int_0^t H_s^2\,ds$ is the accumulated squared magnitude of the integrand. In finance, if $H_t$ is the dollar amount in the risky asset and $dS/S = \sigma\,dW$, then the portfolio quadratic variation is $\int_0^t H_s^2\sigma_s^2\,ds$ — the accumulated squared dollar-volatility exposure.

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2.8 Itô versus Stratonovich

The Itô integral uses the left endpoint $H_k = H_{t_k}$. An alternative is the Stratonovich integral, which uses the midpoint $\frac12(H_{t_k}+H_{t_{k+1}})$:

$$\int_0^T H_t\circ\,dW_t := \lim_{|\Pi|\to 0}\sum_k \frac{H_{t_k}+H_{t_{k+1}}}{2}(W_{t_{k+1}}-W_{t_k}).$$

The two integrals are related by the correction formula

$$\int_0^T H_t\circ\,dW_t = \int_0^T H_t\,dW_t + \tfrac12[H,W]_T,$$

where $[H,W]_T = \int_0^T dH_s\,dW_s$ is the cross-variation.

For a smooth function $f$ applied to Brownian motion, $H_t = f(W_t)$, one computes

$$[f(W),W]_T = \int_0^T f'(W_s)\,ds,$$

so the Stratonovich integral satisfies the classical chain rule: $\int f'(W_t)\circ dW_t = f(W_T)-f(W_0)$. The Itô integral instead satisfies the modified Itô formula in Chapter 3.

Which integral to use?

Itô: natural in finance (adapted integrands, no future look-ahead, martingale property), but its calculus has second-order correction terms.

Stratonovich: natural in physics/geometry (satisfies classical chain rule, coordinate-invariant under manifold transformations), but uses anticipating information.

In this course the default is Itô.

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2.9 The stochastic integral as a process

For fixed $T$, the map $H\mapsto \int_0^T H_t\,dW_t$ is a random variable. But we can also view the integral as a process in the upper limit:

$$M_t = \int_0^t H_s\,dW_s,\qquad 0\le t\le T.$$

A version of $M$ with continuous sample paths exists whenever $H\in\mathcal L^2$.

Continuity and the Burkholder–Davis–Gundy inequality

For $H\in\mathcal L^2$, the martingale $M_t = \int_0^t H_s\,dW_s$ has a continuous version. Moreover, for any $p\ge 1$, there exist universal constants $c_p, C_p$ such that

$$c_p\,\mathbb E\!\left[\left(\int_0^T H_s^2\,ds\right)^{p/2}\right] \le \mathbb E\!\left[\sup_{0\le t\le T}|M_t|^p\right] \le C_p\,\mathbb E\!\left[\left(\int_0^T H_s^2\,ds\right)^{p/2}\right].$$

This inequality controls the supremum of the process in terms of the quadratic variation — a powerful tool in SDE existence theory.

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2.10 A worked example

Compute $\int_0^T W_t\,dW_t$.

Here $H_t = W_t$. This is adapted. On a partition $\{t_k\}$, the Itô sum is

$$I_n = \sum_{k=0}^{n-1} W_{t_k}\,(W_{t_{k+1}}-W_{t_k}).$$

Use the algebraic identity $ab = \frac12[(a+b)^2 - a^2 - b^2]$:

$$W_{t_k}(W_{t_{k+1}}-W_{t_k}) = \tfrac12 W_{t_{k+1}}^2 - \tfrac12 W_{t_k}^2 - \tfrac12(W_{t_{k+1}}-W_{t_k})^2.$$

Summing (telescoping):

$$I_n = \tfrac12 W_T^2 - \tfrac12 W_0^2 - \tfrac12\sum_k(\Delta W_k)^2.$$

Since $W_0 = 0$ and $\sum_k(\Delta W_k)^2\to T$,

$$\int_0^T W_t\,dW_t = \tfrac12 W_T^2 - \tfrac12 T.$$

The correction $-\frac T2$ is the signature of Itô calculus. Classical calculus would give $\int_0^T x\,dx = \frac12 x^2\big|_0^T = \frac12 T$ for $x = T$, but that is the wrong analogy. The Itô integral of $W_t$ against $dW_t$ is not $\frac12 T$; it is the random variable $\frac12 W_T^2 - \frac12 T$.

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2.11 Problems

Problem 1: Itô isometry for a deterministic integrand

Let $H_t = h(t)$ be a deterministic square-integrable function. Compute $\text{Var}\!\left(\int_0^T h(t)\,dW_t\right)$ using the isometry.

Problem 2: Integral of a constant

Let $H_t = c$ (constant). Verify from the definition that $\int_0^T c\,dW_t = c\,W_T$, and check the Itô isometry.

Problem 3: Cross-variation

Let $M_t = \int_0^t H_s\,dW_s$ and $N_t = \int_0^t K_s\,dW_s$ for $H,K\in\mathcal L^2$. Show that

$$[M,N]_T = \int_0^T H_s K_s\,ds.$$

Problem 4: The Stratonovich correction

If $H_t = W_t$, show that the Stratonovich integral satisfies

$$\int_0^T W_t\circ\,dW_t = \tfrac12 W_T^2.$$

Reconcile this with $\int_0^T W_t\,dW_t = \tfrac12 W_T^2 - \tfrac12 T$ via the correction formula.

Problem 5: Exponential martingale

Let $\theta\in\mathbb R$. Show that

$$\mathcal E_t(\theta) = \exp\!\left(\theta W_t - \tfrac12\theta^2 t\right)$$

is a martingale by verifying $d\mathcal E_t = \theta\mathcal E_t\,dW_t$ (use the Itô formula from Chapter 3 in anticipation).

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2.12 Selected solutions

Solution 1

By the Itô isometry,

$$\text{Var}\!\left(\int_0^T h(t)\,dW_t\right) = \mathbb E\!\left[\left(\int_0^T h(t)\,dW_t\right)^2\right] = \int_0^T h(t)^2\,dt.$$

The integral is also Gaussian with mean zero, so the distribution is fully determined by this variance.

Solution 2

For constant $H_t = c$, the simple-process sum is $c\sum_k\Delta W_k = c(W_T - W_0) = cW_T$. The isometry gives $\mathbb E[(cW_T)^2] = c^2 T = c^2\int_0^T dt$. Both sides match.

Solution 3

By polarization: $[M,N] = \frac14([M+N,M+N] - [M-N,M-N])$. The quadratic variation of $\int(H\pm K)\,dW$ is $\int(H\pm K)^2\,ds$. Therefore

$$[M,N]_T = \frac14\int_0^T[(H+K)^2-(H-K)^2]\,ds = \int_0^T H_s K_s\,ds.$$

Solution 4

On each subinterval, the Stratonovich term is

$$\frac{W_{t_k}+W_{t_{k+1}}}{2}\,\Delta W_k = W_{t_k}\,\Delta W_k + \tfrac12(\Delta W_k)^2.$$

Summing and passing to the limit,

$$\int_0^T W_t\circ\,dW_t = \int_0^T W_t\,dW_t + \tfrac12[W]_T = \left(\tfrac12 W_T^2 - \tfrac12 T\right) + \tfrac12 T = \tfrac12 W_T^2,$$

which is the ordinary calculus answer.

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2.13 What this chapter gives you

The Itô integral $\int_0^T H_t\,dW_t$ is:

Property	Statement
Well-defined	for $H\in L^2_{\mathrm{ad}}$ via $L^2$ completion
Zero mean	$\mathbb E[\int H\,dW] = 0$
$L^2$ isometry	$\mathbb E[(\int H\,dW)^2] = \mathbb E[\int H^2\,dt]$
Martingale	$(M_t = \int_0^t H\,dW)$ is a martingale
Quadratic variation	$[M]_t = \int_0^t H_s^2\,ds$
Continuous paths	a continuous version of $M_t$ exists

The next chapter derives Itô's formula, the chain rule of stochastic calculus. Itô's formula is what makes the integral computable and connects it to classical analysis and partial differential equations.