Stochastic Processes and Financial Mathematics
(part two)

12.3 Existence of Ito integrals \(\offsyl \)

This section is off-syllabus, and as such is marked with \(\offsyl \). It will not be covered in lectures.

The argument that proves Theorem 12.2.1, through justifying the limit taken in (12.3), is based heavily on martingales, metric spaces and Hilbert spaces. It comes in two steps, the first of which involves a class of stochastic processes \(F\) known as simple processes – see Definition 12.3.1 below. The second step uses limits extends the definition for simple processes onto a much larger class. We’ll look at these two steps in turn.

We’ll use the notation \(\wedge \) and \(\vee \) from Chapter 8. That is, we write \(\min (s,t)=s\wedge t\) and \(\max (s,t)=s\vee t\).

For a simple process \(F\), with \((t_i)\) as in Definition 12.3.1 we define

\begin{equation} \label {eq:ito_simple} I_F(t)=\sum \limits _{i=1}^n F_{t_{i-1}}[B_{t_i\wedge t}-B_{t_{i-1}\wedge t}]. \end{equation}

Note that this is essentially the right hand side of (12.3) but without the limit. The point of the \(\wedge t\) is that we are aiming to define an integral over \([0,t]\); the \(\wedge t\) makes sure that \(I_F(t)\) only picks up increments from the Brownian motion during \([0,t]\).

We can already see the connection to martingales (which builds on Remark 12.1.1):

Proof: Since a (finite) sum of martingales is a also martingale, it is enough to fix \(i\) and show that \(M_t=F_{t_{i-1}}[B_{t_i\wedge t}-B_{t_{i-1}\wedge t}]\) is a martingale. The argument is rather messy, because we have to handle the \(\wedge t\) everywhere.

Let us look first at \(L^1\). \(F_{t_i}\) is bounded we have some deterministic \(A\in \R \) such that \(|F_{t_i}|\leq A\) (almost surely). Hence, \(\E \l [\l |F_{t_{i-1}}[B_{t_i\wedge t}-B_{t_{i-1}\wedge t}]\r |\r ]\leq A\,\E [|B_{t_i\wedge t}-B_{t_{i-1}\wedge t}|]<\infty .\) Here, we use that \(B_{t_i\wedge t}-B_{t_{i-1}\wedge t}\sim N(0,\,t_i\wedge t-t_{i-1}\wedge t)\), which is in \(L^1\). Hence, \(M_{t}\in L^1\).

Next, adaptedness, for which we consider two cases.

• • If \(t\geq t_{i-1}\) then \(F_{t_{i-1}}\in \mc {F}_t\). Since \(t_i\wedge t\leq t\), we have \(B_{t_i\wedge t}\in m\mc {F}_t\) and, similarly, \(B_{t_{i-1}\wedge t}\in m\mc {F}_t\), hence also \(M_t\in m\mc {F}_t\).

• • If \(t<t_{i-1}\) then \(t_i\wedge t=t_{i-1}\wedge t=t\), meaning that \(B_{t_i\wedge t}-B_{t_{i-1}\wedge t}=0\). So \(M^{(i)}_{t_i}=0\), which is deterministic and therefore also in \(m\mc {F}_t\).

Therefore, \((M_t)\) is adapted to \((\mc {F}_t)\).

Lastly, let \(0\leq u\leq t\). Again, we consider two cases.

• • If \(u\geq t_{i-1}\) then \(F_{t_{i-1}}\in \mc {F}_u\) and we have

  \(\seteqnumber{0}{12.}{9}\)

  \begin{align*} \E \l [F_{t_{i-1}}[B_{t_i\wedge t}-B_{t_{i-1}\wedge t}]\|\mc {F}_u\r ] &=F_{t_{i-1}}\l (\E \l [B_{t_i\wedge t}\|\mc {F}_u\r ]-\E \l [B_{t_{i-1}\wedge t}\|\mc {F}_u\r ]\r )\\ &=F_{t_{i-1}}[B_{t_i\wedge t \wedge u}-B_{t_{i-1}\wedge t\wedge u}]\\ &=F_{t_{i-1}}[B_{t_i\wedge u}-B_{t_{i-1}\wedge u}]. \end{align*} Here, in the first line we take out what is known, and we use the martingale property of Brownian motion to deduce the second line. The third line then follows because \(u\leq t\).

• • If \(u< t_{i-1}\) then \(B_{t_i\wedge u}-B_{t_{i-1}\wedge u}=0\). Also, by the tower rule

  \(\seteqnumber{0}{12.}{9}\)

  \begin{align*} \E \l [F_{t_{i-1}}[B_{t_i\wedge t}-B_{t_{i-1}\wedge t}]\|\mc {F}_u\r ] &=\E \l [\E \l [F_{t_{i-1}}[B_{t_i\wedge t}-B_{t_{i-1}\wedge t}]\|\mc {F}_{t_{i-1}}\r ]\|\mc {F}_u\r ]\\ &=\E \l [F_{t_{i-1}}\l (\E \l [B_{t_i\wedge t}\|\mc {F}_{t_{i-1}}\r ]-\E \l [B_{t_{i-1}\wedge t}\|\mc {F}_{t_{i-1}}\r ]\r )\|\mc {F}_u\r ]\\ &=\E \l [F_{t_{i-1}}\l (B_{t_i\wedge t\wedge t_{i-1}}-B_{t_{i-1}\wedge t\wedge t_{i-1}}\r )\|\mc {F}_u\r ]\\ &=\E \l [F_{t_{i-1}}\l (B_{t_{i-1}}-B_{t_{i-1}}\r )\|\mc {F}_u\r ]\\ &=0. \end{align*}

In both cases, we have shown that \(\E [M_t\|\mc {F}_u]=M_u\).   ∎

Proof: See exercise 12.10. The proof similar in style to that of Lemma 12.3.2.   ∎

Essentially, Theorem 12.2.1 says that Ito integrals exist for \(F\in \mc {H}^2\) and that Lemmas 12.3.2 and 12.3.3 are true, not just for simple processes, but for Ito integrals in general. This observation brings us to second step of the construction of Ito integrals, although we won’t be able to cover all of the details here. It comes in two sub-steps:

• 1. Fix \(t<\infty \) and begin with a process \(F\in \mc {H}^2\). Approximate \(F\) by a sequence of simple processes \(F^{(k)}\) such that

  \(\seteqnumber{0}{12.}{10}\)

  \begin{equation} \label {eq:ito_approx_cond} \int _0^t\E \l [\l (F_u-F^{(k)}_u\r )^2\r ]\,du\to 0 \end{equation}

  as \(k\to \infty \). It can be proved that this is always possible.

• 2. For each \(k\), \(I_{F_m}(t)\) is defined by (12.9). We define

  \(\seteqnumber{0}{12.}{11}\)

  \begin{equation} \label {eq:ito_simple_limit} \int _0^t F_u\,dB_u=\lim \limits _{k\to \infty }I_{F^{(k)}}(t). \end{equation}

  Using (12.11), it can be shown that this limit exists, with convergence in \(L^2\), and moreover its value (on the left hand side) is independent of the choice of approximating sequence \(F^{(k)}\) (on the right hand side).

We end with a brief summary of the mathematics that lies behind (12.11) and (12.12). We have shown that the map \(F\mapsto I_F\) takes a simple process, which is an example of a locally square integrable adapted stochastic process, and gives back a martingale that is in \(\mc {L}^2\). If we add appropriate restrictions on the left and right continuity of \(F\), it can be shown that the map \(F\mapsto I_F\) becomes a linear operator between two Hilbert spaces. Further, (12.10) turns out to be precisely the statement that \(F\mapsto I_F\) is an isometry (usually referred to as the Ito isometry). The set of simple stochastic processes is a dense subset of the space of square integrable adapted stochastic processes, which allows us to use a powerful theorem about isometries between Hilbert spaces (known as the completion theorem) to take the limit in (12.12).