Three line summary

The Itô integral is a way of integrating random variables against Brownian motion.
The Itô integral is well defined for piecewise constant adapted processes $\mathcal{E}$ and turns them isometrically into square integrable continuous martingales ($\mathcal{M}\U I^2$).
As a result the Itô integral can be extended isometrically to a function $\overline{\mathcal{E}}\to \mathcal{M}\U I^2$. Furthermore $\overline{\mathcal{E}}$ can be characterised explicitly as the square integrable adapted processes that are measurable in time and space.

Why should I care?

The Itô integral forms the basis of the whole of stochastic calculus. This comprises SDEs, SPDEs. Knowledge of what functions can be integrated and what properties the integrated function has is instrumental. In this post we construct the integral and address both of the preceding issues.

Notation

Given two measure spaces $(\Omega,\mathcal{F}),(\Omega’,\mathcal{H})$ we abbreviate that $f:\Omega\to\Omega’$ is measurable between $\mathcal{F}$ and $\mathcal{F}’$ as $f:\mathcal{F}\to\mathcal{F}’.$ Furthermore we will take $I=\zl 0,T\zr $ or $I=\zl 0,+\infty\zr$ to be the index set of our stochastic processes and write $\mathcal{F}\U \infty:=\vee\U {t\in I}\mathcal{F}\U t$. Finally, we will denote the Borel $\sigma$-algebra on some interval $J$ by $\mathcal{B}(J)$.

Integrable functions: progressive measurability

As we will soon see the only stochastic process that can be integrated are the square integrable and progressively measurable ones. But what does this mysterious term mean?

Definition 1. A stochastic process $\{ X\U t \}\U {t\in I}$ is progressively measurable if

$$X:\mathcal{B}(\zl 0,t\zr )\otimes\mathcal{F}\U t\to \mathcal{H}$$

is measurable for all $t\in I$.

Whenever we’re given a stochastic process and a filtration the first thing to check is that it is adapted. In fact, since $\omega\to(t,\omega)$ is $\mathcal{F}\U t\to \mathcal{B}(\zl 0,t\zr )\otimes \mathcal{F}\U t$ measurable for all $t$ we have that the following holds.

Lemma 1. Progressively measurable processes are adapted.

Additionally, stochastic processes can be viewed path-wise but also be seen as functions of a product space, this leads to the following definition.

Definition 2. We say that a stochastic process $\{X\U t\}\U {t\in I}$ is jointly measurable if

$$X:\mathcal{B}(I)\otimes\mathcal{F}\U \infty\to \mathcal{H}$$

In the definition of progressive measurability we imposed some kind of measurability, in fact the condition leads to the following

Proposition 1 (Progressive implies jointly measurable). Let $I\subset{\mathbb R}$, and $\{X\U t\}\U {t\in I}$ be progressively measurable. Then it is also jointly measurable.

Proof. Given $A\in\mathcal{H}$ we have that

$$\begin{gathered} \left.{X^{-1}}\right|\U {\zl 0,t\zr \times\Omega}(A)\in\mathcal{B}(\zl 0,t\zr )\otimes\mathcal{F}\U t\quad\forall t\in I\\ \iff X^{-1}(A)\cap(\zl 0,t\zr \times\Omega)\in\mathcal{B}(\zl 0,t\zr )\otimes\mathcal{F}\U t\quad\forall t\in I\\\implies X^{-1}(A)=\bigcup\U {n \in \mathbb{N}}X^{-1}(A)\cap(\zl 0,t\U n\zr \times\Omega) \in\mathcal{B}(I)\otimes\mathcal{F}\U \infty \end{gathered}$$

Where $t\U n\in I$ is a sequence converging to the endpoint of $I$. ◻

Note however that the converse isn’t true, for example if $X$ is constant in $t$ then, for some $B\subset \Omega$ it holds that

$$X^{-1}(A)=I\times B;\quad {X^{-1}}|\U {\zl 0,t\zr \times\Omega}(A)= \zl 0,t\zr \times B$$

So it suffices to consider some construction where $B\in\mathcal{F}\U \infty$ but $B\not\in\mathcal{F}\U t$.

The difference between progressively measurable and adapted is quite subtle. In fact every adapted and jointly measurable stochastic process has a progressively measurable modification (see 1 page $5$). The proof of this fact is very lengthy and technical so we ommit it. However, this shows that if $X\in L^2(I\times\Omega,\mathcal{B}(I)\otimes\mathcal{F}\U \infty)$ (and in particular $X$ is jointly measurable), we may always choose a representative that is progressively measurable. This leads to some authors giving the definition of the class of Itô integrable functions in terms of joint measurability instead of progressive measurability. In the end both lead to equivalent definition. That said, this technicality is usually of little importance due to the following result.

Lemma 3 (Continuity, when adapated, is progressive). Let $\{X\U t\}\U {t\in I}$ be a left or right continuous adapted stochastic process. Then $X$ is progressively measurable.

Proof. Suppose for example that $X$ is right continuous. Let us fix $t \in I$ and set $X^{(n)}_0=X_0$ and for $k=1,…,2^{n}-1$

$$X\U {s}^{(n)}(\omega)=X\U {(k+1)t / 2^{n}}(\omega) \text { for } \frac{k t}{2^{n}}<s \leq \frac{k+1}{2^{n}} t$$

Since $X^{(n)}$ is piecewise constant in time, the pre-image of any set $A\in\mathcal{H}$ is of the form

$$(X^{(n)})^{-1}(A)=\bigcup\U {i}(t^n\U i,t^n\U {i+1}) \times X\U {t^n\U {i+1}}^{-1}(A),$$

for some $t_i^n \in [0,t]$. So $X^{(n)}$ is $\mathcal{B}(\zl 0, t \zr\times \Omega)$ measurable. Furthermore by right continuity $\lim\U {n \to \infty}X^{(n)}=X$. So $X$ is also $\mathcal{B}(\zl 0, t \zr\times \Omega)$ measurable. Since $t$ was any we conclude. ◻

Later on we will see that the solutions to an SDE (which are defined by Itô integration) are continuous, and thus progressively measurable. This motivates the title of the following lemma.

Lemma 2 (SDE coefficients are progressive). Let $X\U t$ be a progressively measurable stochastic process and let $f:\mathcal{H}\to\mathcal{G}$ be measurable, then $f(t,X\U t)$ is progressively measurable.

Proof. This follows from considering $(t,\omega)\to (t,X(t,w))$. Where the arrow is measurable as, due to the progressive measurability of $X$, each component is adapted. ◻

We now move to defining the Itô integral. We will first define it on step processes of the form

\[S(t)=X\U 0 1\U 0(t)+\sum\U {n=0}^{N-1} S(t\U n)1\U{ [t\U n,t\U{n+1})}\]

where $S \U n$ are $\mathcal{F}\U {t \U n}$ measurable.

Lemma 4. For any $p \in\zl 1, \infty)$, the simple processes are $L^{p}$-dense in the space $\mathbb{L}^{p}(I\times\Omega)$ of progressively measurable processes in $L^p(\mathcal{B}(I)\otimes\mathcal{F}\U \infty)$. That is, for any $Y \in \mathbb{L}^{p}$ there is a sequence $S\U {n}$ of elementary functions such that

$$\mathbb{E}\left [ \int\U {I}|Y(t)-S\U {n}(t)|^{p} dt\right ] \longrightarrow 0.$$

The proof of this fact is also rather technical and long. See Chapter $2$ of 2. Furthermore, we have that

Theorem 1. Let $(\mathcal{E},| \cdot |\U {L^2(I\times\Omega)},\mathcal{F}\U t)$ be the set of simple stochastic processes adapted to $\{\mathcal{F}\U t\}\U {t\in I}$ with the $L^2$ norm. Then it’s completion is

$$\mathbb{L}^2(I\times\Omega):=\\{X\in L^2(\zl 0,T\zr \times\Omega)\text{ progressively measurable }\\}.$$

The proof of this is by the previous approximation result together with the fact that the Ito integral of simple processes is an isometry and the fact that $\mathbb{L}^2(I\times\Omega)$ is complete. This last property follows from the completeness of the $L^p$ spaces and the fact that pointwise limits of progressively measurable functions are progressively measurable (which we can use as from every convergent sequence in $L^p$ we can extract a convergent sub-sequence which must also converge to the $L^p$ limit). This finally leads us to be able to define the stochastic integral.

Theorem 2. Let $t\in I$ and define for a simple process $f\in\mathcal{E}$

$$\int\U {0}^t X dW=\sum\U {n=0}^{N-1} X(t\U n)(W(t\cap t\U {n+1})-W(t\U n)).$$

Then the above defines an isometry to the space of continuous square integrable martingales $\mathcal{M}\U I^2$ as

$$\begin{aligned} int: \left(\mathcal{E},\norm{\cdot}\U {L^2(I\times\Omega)}\right) & \longrightarrow \left(\mathcal{M}\U I^2,\norm{\cdot}\U {L^2(I\times\Omega)}\right) \\ X(t) & \longmapsto \int\U {0}^t X dW .\end{aligned}$$

Thus, it extends uniquely to the closure $\overline{\mathcal{E}}=\mathbb{L}^2(I\times\Omega)$. Furthermore the extension also has image in $\mathcal{M}\U I^2.$

Proof. Proving this holds for simple processes is a calculation using: the definition of the Itô integral of a simple process, the adaptedness of $X$ and the definition of $W$. The fact that the extension of the integral is also in $\mathcal{M}\U I^2$ is due to the fact that, by Doob’s maximal inequality and the completeness of $L^p$ spaces, $\mathcal{M}\U I^2$ is a Banach space (in fact it’s a Hilbert space). In fact, this holds even for processes valued in other separable Banach spaces (and not just $\mathbb{R}$), but this is a matter for another post ◻

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Tags: Stochastic calculus