Stochastic Processes and Financial Mathematics
(part two)

12.2 Ito integrals

In Section 12.1 we discussed the ideas behind Ito integrals. We did not discuss one key (theoretical) question: if and when the limit in (12.3) actually exists?

Let us recall our usual notation. We work over a filtered space $(\mathrm{\Omega },\mathcal{F},({\mathcal{F}}_t),\mathbb{P})$, where the filtration ${\mathcal{F}}_t$ is the generated filtration of a Brownian motion $B_t$. We use the letters $t,u$ and sometimes also $v$, as our time variables.

We say that a stochastic process $F_t$ is locally square integrable if

$$\begin{array}{}\text{(12.5)}& {\int }_0^t\mathbb{E}\left[F_u^2\right]du<\mathrm{\infty }.\end{array}$$

for all $t\in [0,\mathrm{\infty })$. We define ${\mathcal{H}}^2$ to be the set of locally square integrable continuous stochastic processes $F=(F_t{)}_{t=0}^{\mathrm{\infty }}$ that are adapted to $({\mathcal{F}}_t)$.

It turns out that the condition $F\in {\mathcal{H}}^2$ is the correct condition under which to take the limits discussed in Section 12.1. The following theorem formally states that Ito integrals exist, and gives some of their first properties.

So far we have only looked at integrals over $[0,t]$. We can extend the definition to ${\int }_a^b$, simply by repeating the whole procedure above with limits $[a,b]$ instead of $[0,t]$. It is easily seen that this gives the usual consistency property

$$\begin{array}{}\text{(12.6)}& {\int }_a^c{F}_{t}d{B}_t={\int }_a^b{F}_{t}d{B}_t+{\int }_b^c{F}_{t}d{B}_t\end{array}$$

for $a\le b\le c$. We won’t include a proof of this in our course.

Like classical integrals, Ito integrals are linear. For $\alpha ,\beta \in \mathbb{R}$ we have

$$\begin{array}{}\text{(12.7)}& {\int }_a^b\alpha F_t+\beta G_{t}d{B}_t=\alpha {\int }_a^b{F}_{t}d{B}_t+\beta {\int }_a^b{G}_{t}d{B}_t.\end{array}$$

Again, we won’t include a proof of this formula in our course.

In future, we’ll use the linearity and consistency properties without comment. However, as we’ll explore in the next two sections, there are many ways in which the Ito integral does not behave like the classical integral.

Comparing Ito integration to classical integration

Let us first note one similarity. It is true that

$${\int }_0^{t}0d{B}_u=0.$$

This matches classical integrals, where we have ${\int }_0^{t}0du=0$. We can see this from (12.3), by setting $f\equiv 0$, and noting that the limit of $0$ is $0$.

Here’s a first difference: fix some $t>0$ and let us look at ${\int }_0^{t}1d{B}_u$. If we set $f\equiv 1$ in (12.3), then we obtain $\sum _{i=1}^{n}f(t_{i-1})[B_{t_i}-B_{t_{i-1}}]=B_t-B_0=B_t$, and hence

$$\begin{array}{}\text{(12.8)}& {\int }_0^{t}1d{B}_u=B_t.\end{array}$$

Of course, in classical calculus we have ${\int }_0^{t}1du=t$. This is our first example of an important principle: Ito integration behaves differently to classical integration. To illustrate further, in Section 13.1 we will put together a set of tools for calculating Ito integrals, and in Example 13.1.2 we will see that

$${\int }_0^t{B}_{u}d{B}_u=\frac{B_t^2}{2}-\frac{t}{2}.$$

This corresponds to taking $f(t)=B_t$ in (12.3). Of course, in classical calculus we have ${\int }_0^{t}udu=\frac{u^2}{2}$, which is very different.