# Motivation behind the definition of the Itô integral for elementary predictable processes

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space and $\mathbb{F}$ be a filtration on $(\Omega,\mathcal{A})$. A real-valued stochastic process $H=(H_t)_{t\ge 0}$ is called elementary $\mathbb{F}$-predictable $:\Leftrightarrow$ $H$ is $\mathbb{F}$-adapted, locally bounded and of the form $$H_t(\omega)=\sum_{i=1}^nH_{t_{i-1}}(\omega)1_{(t_{i-1},t_i]}(t)\;\;\;\text{for all }(\omega,t)\in\Omega\times [0,\infty)\;.$$

Now, the Itô-Integral of $H$ with respect to a Brownian motion $B$ is defined as $$I_\infty^B(H):=\sum_{i=1}^nH_{t_{i-1}}(B_{t_i}-B_{t_{i-1}})$$

One can show, that $H\in\mathcal{L}^2\left(\operatorname{P}\otimes\left.\lambda^1\right|_{[0,\infty)}\right)$ and that the Itô isometry is satisfied.

However, why should this be of interest? Clearly, we expect some things like linearity from an integral. But I really don't understand, why it is important, that the space of such $H$ is a subspace of $\mathcal{L}^2\left(\operatorname{P}\otimes\left.\lambda^1\right|_{[0,\infty)}\right)$ and why the Itô isometry is so important.

## 1 Answer

The Itô isometry is important because it allows you to extend the stochastic integrals that aren't in general path-by-path well defined to any function in $L^2$.

Remember that the Brownian motion is of unbounded variation and the integral in Stieltjes sense is not readily available. If you would like to see a path-by-path construction of the integral you should read this introductory notes http://ir.nmu.org.ua/bitstream/handle/123456789/129023/2c877e24f323a77e5c8e3a120d5105ac.pdf?sequence=1

• What is meant by path-by-path? – 0xbadf00d Jul 4 '15 at 10:12
• fix a path of the brownian motion $\omega$, now integrate with $f$ with respect to this path. (if this continuous path is ugly then you can't integrate $f$ with respect to $\omega$). Ito isometry is a way to make sense of the integral for those bad trajectories. – Conrado Costa Jul 6 '15 at 1:14