The Itô Integral

The Itô integral extends classical integration to allow integrands that are stochastic processes and integrators that are Brownian motion. It is the foundation of stochastic calculus.

Motivation

Classical Riemann-Stieltjes integration $\int_0^T f(t) \, dg(t)$ requires $g$ to have bounded variation. But Brownian motion has infinite variation, so we cannot define $\int_0^T f(t) \, dB_t$ using classical methods. Itô (1944) developed a new theory based on the finite quadratic variation of Brownian motion.

Definition for simple processes

Definition5.1Simple process

A simple process is $H_t = \sum_{i=0}^{n-1} H_i \mathbf{1}_{[t_i, t_{i+1})}(t)$ , where $0 = t_0 < t_1 < \cdots < t_n = T$ and each $H_i$ is $\mathcal{F}_{t_i}$ -measurable with $\mathbb{E}[H_i^2] < \infty$ .

For simple $H$ , define:

$I_T(H) = \int_0^T H_t \, dB_t := \sum_{i=0}^{n-1} H_i (B_{t_{i+1}} - B_{t_i}).$

Theorem5.1Properties of I(H)

$I_T(H)$ is $\mathcal{F}_T$ -measurable.
$\mathbb{E}[I_T(H)] = 0$ .
Itô isometry: $\mathbb{E}[I_T(H)^2] = \mathbb{E}\left[\int_0^T H_t^2 \, dt\right]$ .

The Itô isometry is the key: it shows that the Itô integral is an $L^2$ isometry from integrands to integrals.

Extension to general integrands

Definition5.2Itô integrable

A process $(H_t)$ is Itô integrable if:

$H_t$ is adapted to the filtration of $B$ .
$\mathbb{E}\left[\int_0^T H_t^2 \, dt\right] < \infty$ .

The space of Itô integrable processes is $L^2([0, T] \times \Omega)$ with respect to the measure $dt \times d\mathbb{P}$ .

By approximating general $H$ with simple processes and using the Itô isometry, we extend the definition to all Itô integrable processes.

Theorem5.2Existence and uniqueness

For any Itô integrable $H$ , there exists a unique (up to indistinguishability) process $(I_t(H))_{0 \leq t \leq T}$ such that:

$(I_t(H))$ is a continuous martingale.
$I_0(H) = 0$ .
$[I(H)]_T = \int_0^T H_t^2 \, dt$ (quadratic variation).

Properties

Theorem5.3Martingale property

The Itô integral $M_t = \int_0^t H_s \, dB_s$ is a continuous local martingale with $\mathbb{E}[M_t] = 0$ and $\text{Var}(M_t) = \mathbb{E}\left[\int_0^t H_s^2 \, ds\right]$ .

ExampleIntegrating a constant

$\int_0^t 1 \, dB_s = B_t$ . The integral of the constant function 1 is Brownian motion itself.

ExampleIntegrating Brownian motion

$\int_0^t B_s \, dB_s = \frac{1}{2}B_t^2 - \frac{1}{2}t$ . Note the correction term $-t/2$ — this is the Itô correction, arising from the quadratic variation of $B$ .

Summary

The Itô integral $\int_0^t H_s \, dB_s$ is defined via $L^2$ approximation, satisfies the Itô isometry $\mathbb{E}[(\int H \, dB)^2] = \mathbb{E}[\int H^2 \, dt]$ , and is a martingale. It enables stochastic differential equations and stochastic calculus.