Itô integral

The Itô integral

\int_{0}^{t} H_{s} d W_{s}

integrates a predictable square-integrable process against Brownian motion. It is defined first for predictable step processes and then completed using the Itô isometry.

Table of contents

Itô integrand of a finite sum of Gaussian exponentials ( $f_{t} = \sum_{j} c_{j} h_{j} (t) exp (\int_{0}^{t} h_{j} d B - \frac{1}{2} \int_{0}^{t} h_{j}^{2} d s)$ )

 0  0

For deterministic continuous functions

h_{j}

on a finite interval, their stochastic exponentials

M (h_{j})

are square-integrable martingales and satisfy

d M (h_{j}) = h_{j} M (h_{j}) d B

. Thus

F = \sum_{j} c_{j} M (h_{j})_{T}

has mean

\sum_{j} c_{j}

and stochastic-integral representation with the displayed integrand. Itô isometry proves uniqueness up to time-probability null sets among square-integrable predictable integrands. It does not give pointwise uniqueness, since changing a value at one deterministic time changes no integral.

Iterated Itô integral ( $I_{n} (t) = \int_{0}^{t} I_{n - 1} (s) d B_{s}, I_{0} = 1$ )

 0  0

For Brownian motion, an ordered iterated Itô integral is defined by the displayed recursion. It represents integration on

0 \leq s_{1} \leq \dots \leq s_{n} \leq t

, with time increasing toward the outermost Itô integral. The Itô isometry proves inductively that

E I_{n} (t)^{2} = \int_{0}^{t} E I_{n - 1} (s)^{2} d s = \frac{t ^{n}}{n !} .

(1)

Thus every recursive integrand is a square-integrable stochastic integrand on each finite interval, and

I_{n}

is a martingale for

n \geq 1

. For

n = 1, 2, 3

, increment algebra gives

I_{1} = B_{t}

I_{2} = (B_{t}^{2} - t) /2

and

I_{3} = (B_{t}^{3} - 3 t B_{t}) /6

. These are the corresponding space-time Hermite polynomials divided by

n!

Gaussianity of deterministic Brownian stochastic integrals ( $\int_{0}^{t} h (s) d B_{s} \sim N (0, \int_{0}^{t} h (s)^{2} d s)$ )

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For deterministic square-integrable

h

, approximate by step functions. Their Itô integrals are sums of independent normal Brownian increments. The Itô isometry gives convergence in

L^{2}

, and the variances converge to

\int h^{2}

; convergence of characteristic functions proves the normal limit. Each linear combination of finitely many such integrals is another deterministic stochastic integral, so they are jointly Gaussian.

Gaussian coordinates of deterministic orthonormal Wiener integrands ( $ξ_{n} = \int g_{n} d W independent N (0, 1)$ )

 0  0

Deterministic Itô integrals are jointly centered Gaussian by approximation from step functions and the Itô isometry. Their covariance is the L2 inner product of their integrands. Orthonormal integrands therefore produce independent standard-normal coordinates, with independence of a countable family understood through all finite subfamilies.