The Ito isometry

The Itō isometry is a useful theorem in stochastic calculus that provides a fundamental tool in computing stochastic integrals - integrals with respect to a Brownian motion \begin{equation} \int_{0}^{\infty} f(s) dB_{s} \end{equation} with $B_{s}$ a Brownian motion.

First, we'll define a predictable process.

Definition

A stochastic process $\alpha_{t}$ is called a simple predictable process if it is of the form \begin{equation} \alpha_{t} = \sum_{i=1}^{n} \mathbb{I}_{(t _{i-1}, t_{i}]} a_{i} \end{equation} for $0 \leq t_{0} < \dots < t_{n}$, with $a_{i}$ a bounded $\mathcal{F}_{t_{i-1}}$-measurable random variable.

This definition is useful as we can construct sequences of simple predictable processes that converge in $L^{2}$ to our stochastic processes of interest (under certain technical conditions).

The Itō isometry is the following result

Theorem

Let $\alpha_{t}$ be a simple predictable process. Then

\begin{equation} \mathbb{E} \left(\int_{0}^{\infty} \alpha_{s} dB_{s} \right)^{2} = \mathbb{E} \int_{0}^{\infty} \alpha_{s}^{2} ds \end{equation}

Thus, the mapping from simple predictable process to square integrable random variables on $L^{2}(\Omega, \mathcal{F}, \mathbb{P})$ (which is complete) defined by \begin{equation} I(\alpha) = \int_{0}^{\infty} \alpha_{s} dW_{s} \end{equation} is an isometry.

Proof

The proof is a fairly straightforward application of the properties of the Brownian motion.

The expectation of the square of $\alpha_{t}$ becomes a sum of expectations on disjoint subsets of $[0, \infty)$ (which are zero by the independence of Brownian motion in two intervals), and the sum of the expectation of $a_{i}^{2} (dB_{s})^{2}$ on the interval $(t_{i-1}, t_{i}]$, which is $a_{i}^{2} (t_{i} - t_{i-1})$ as the increments of the Brownian motion are normally distributed with mean zero and variance equal to the size of the interval.

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