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 \((\Omega ,\mc {F},(\mc {F}_t),\P )\), where the filtration \(\mc {F}_t\) 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
\(\seteqnumber{0}{12.}{4}\)\begin{equation} \label {eq:locally_sqr_int} \int _0^t \E \l [F_u^2\r ]\,du <\infty . \end{equation}
for all \(t\in [0,\infty )\). We define \(\mc {H}^2\) to be the set of locally square integrable continuous stochastic processes \(F=(F_t)_{t=0}^\infty \) that are adapted to \((\mc {F}_t)\).
It turns out that the condition \(F\in \mc {H}^2\) 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.
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Theorem 12.2.1 For any \(F\in \mc {H}^2\), and any \(t\in [0,\infty )\) the Ito integral
\[\int _0^t F_u\,dB_u\]
exists, and is a continuous martingale with mean and variance given by
\(\seteqnumber{0}{12.}{5}\)\begin{align*} \E \l [\int _0^t F_u\,dB_u\r ]&=0,\\ \E \l [\l (\int _0^t F_u\,dB_u\r )^2\r ]&=\int _0^t\E [F_u^2]\,du. \end{align*}
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
\(\seteqnumber{0}{12.}{5}\)\begin{equation} \label {eq:intabc} \int _a^cF_t\,dB_t=\int _a^b F_t\,dB_t+\int _b^c F_t\,dB_t \end{equation}
for \(a\leq b\leq c\). We won’t include a proof of this in our course.
Like classical integrals, Ito integrals are linear. For \(\alpha ,\beta \in \R \) we have
\(\seteqnumber{0}{12.}{6}\)\begin{equation} \label {eq:itolinear} \int _a^b \alpha F_t+\beta G_t\,dB_t=\alpha \int _a^b F_t\,dB_t+\beta \int _a^b G_t\,dB_t. \end{equation}
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
\[\int _0^t0\,dB_u=0.\]
This matches classical integrals, where we have \(\int _0^t 0\,du=0\). 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 \(\int _0^t 1\,dB_u\). 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
\(\seteqnumber{0}{12.}{7}\)\begin{equation} \label {eq:intBt} \int _0^t 1\,dB_u=B_t. \end{equation}
Of course, in classical calculus we have \(\int _0^t 1\,du=t\). 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
\[\int _0^t B_u\,dB_u=\frac {B_t^2}{2}-\frac {t}{2}.\]
This corresponds to taking \(f(t)=B_t\) in (12.3). Of course, in classical calculus we have \(\int _0^t u\,du=\frac {u^2}{2}\), which is very different.