Itô Integral
1 Motivation
The foundation of Stochastic Calculus is integration with respect to diffusions, a class of continuous-time Markov processes with almost surely continuous sample paths (see Itô Diffusion). Several constructions exist, notably those of Itô and Stratonovich. They differ in where the integrand is evaluated on each small interval: the Itô integral uses the left endpoint, which keeps the integral a martingale and is the convention in finance. Brownian paths have infinite variation, so the integral cannot be defined pathwise as a Riemann-Stieltjes integral, which is why a separate construction is needed.
Let \(B\) be a (one-dimensional) Brownian motion on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t),\mathbb{P})\) (see Filtrations and Adapted Processes). A diffusion has the form
\[ X_{t+s}-X_t=\int_t^{t+s}\mu(X_u,u)\,du+\int_t^{t+s}\sigma(X_u,u)\,dB_u, \]
and the second term is an Itô integral of the process \(\sigma(X_u,u)\).
2 The Class of Integrands
\(\mathcal{V}=\mathcal{V}(S,T)\) is the class of functions \(f(t,\omega):[0,\infty)\times\Omega\to\mathbb{R}\) such that
- \((t,\omega)\mapsto f(t,\omega)\) is \(\mathcal{B}\times\mathcal{F}\)-measurable, where \(\mathcal{B}\) is the Borel \(\sigma\)-algebra on \([0,\infty)\),
- \(f(t,\cdot)\) is \(\mathcal{F}_t\)-adapted, and
- \(\mathbb{E}\left[\int_S^Tf(t,\omega)^2\,dt\right]<\infty\).
The aim is to define
\[ \mathcal{I}[f](\omega)=\int_S^Tf(t,\omega)\,dB_t(\omega),\qquad f\in\mathcal{V}. \]
We first define it for step functions and then extend by approximation.
3 Elementary Functions
A function \(\phi\in\mathcal{V}\) is elementary if
\[ \phi(t,\omega)=\sum_je_j(\omega)\,\mathbb{1}_{[t_j,t_{j+1})}(t). \]
Since \(\phi\in\mathcal{V}\), each \(e_j\) must be \(\mathcal{F}_{t_j}\)-measurable. For such \(\phi\) define
\[ \int_S^T\phi(t,\omega)\,dB_t(\omega)=\sum_{j\ge 0}e_j(\omega) \left[B_{t_{j+1}}-B_{t_j}\right](\omega). \]
If \(\phi\) is bounded and elementary, then
\[ \mathbb{E}\left[\left(\int_S^T\phi\,dB_t\right)^2\right] =\mathbb{E}\left[\int_S^T\phi^2\,dt\right]. \]
Proof. Write \(\Delta B_j=B_{t_{j+1}}-B_{t_j}\). By the independent increments of Brownian motion, and since \(e_i,e_j\) are measurable with respect to the past,
\[ \mathbb{E}[e_ie_j\Delta B_i\Delta B_j]= \begin{cases} 0 & i\neq j,\\ \mathbb{E}[e_j^2]\,(t_{j+1}-t_j) & i=j. \end{cases} \]
Hence
\[ \mathbb{E}\left[\left(\int_S^T\phi\,dB\right)^2\right] =\sum_{i,j}\mathbb{E}[e_ie_j\Delta B_i\Delta B_j] =\sum_j\mathbb{E}[e_j^2](t_{j+1}-t_j) =\mathbb{E}\left[\int_S^T\phi^2\,dt\right]. \]
4 Extension to \(\mathcal{V}\)
The extension takes three steps, each showing that a larger class can be approximated in \(L^2(\mathbb{P}\times dt)\) by a smaller one.
Step 1. Let \(g\in\mathcal{V}\) be bounded with \(g(\cdot,\omega)\) continuous for each \(\omega\). Then there are elementary \(\phi_n\in\mathcal{V}\) with
\[ \mathbb{E}\left[\int_S^T(g-\phi_n)^2\,dt\right]\to 0. \]
Proof. Take \(\phi_n=\sum_jg(t_j,\omega)\mathbb{1}_{[t_j,t_{j+1})}\) on a partition with mesh tending to \(0\). It is elementary since \(g\) is adapted, and by continuity of \(g(\cdot,\omega)\) we have \(\int_S^T(g-\phi_n)^2dt\to 0\) for each \(\omega\). The claim follows by bounded convergence.
Step 2. Let \(h\in\mathcal{V}\) be bounded. Then there are bounded \(g_n\in\mathcal{V}\), continuous in \(t\) for each \(\omega\), with \(\mathbb{E}\left[\int_S^T(h-g_n)^2dt\right]\to 0\).
Proof. Let \(\psi_n\) be a non-negative continuous function supported in \([-1/n,0]\) with \(\int\psi_n=1\), and set \(g_n(t,\omega)=\int_0^t\psi_n(s-t)h(s,\omega)\,ds\). Then \(g_n\) is continuous in \(t\), bounded, and adapted since it only uses values of \(h\) up to time \(t\). For each \(\omega\), \(\int_S^T(g_n-h)^2dt\to 0\) because \(\psi_n\) is an approximate identity, and bounded convergence gives the result.
Step 3. Let \(f\in\mathcal{V}\). Then there are bounded \(h_n\in\mathcal{V}\) with \(\mathbb{E}\left[\int_S^T(f-h_n)^2dt\right]\to 0\).
Proof. Truncate, \(h_n=\max(-n,\min(f,n))\). Then \(|f-h_n|\le|f|\), so dominated convergence applies since \(\mathbb{E}\int f^2<\infty\).
Combining the steps, every \(f\in\mathcal{V}\) admits elementary \(\phi_n\) with \(\mathbb{E}\left[\int_S^T(f-\phi_n)^2dt\right]\to 0\).
For \(f\in\mathcal{V}(S,T)\) the Itô integral is
\[ \int_S^Tf\,dB_t=\lim_{n\to\infty}\int_S^T\phi_n\,dB_t\quad\text{in } L^2(\mathbb{P}), \]
for any sequence of elementary \(\phi_n\) with \(\mathbb{E}\left[\int_S^T(f-\phi_n)^2dt\right]\to 0\).
The limit exists because, by the isometry applied to \(\phi_n-\phi_m\), the sequence \(\int\phi_n\,dB\) is Cauchy in \(L^2(\mathbb{P})\), and it does not depend on the chosen sequence for the same reason.
5 Itô Isometry
The isometry passes to the limit, which gives the main result for Itô integrals.
For all \(f\in\mathcal{V}(S,T)\),
\[ \mathbb{E}\left[\left(\int_S^Tf(t,\omega)\,dB_t\right)^2\right] =\mathbb{E}\left[\int_S^Tf^2(t,\omega)\,dt\right]. \]
6 Properties
Let \(f,g\in\mathcal{V}(0,T)\) and \(0\le S<U<T\). Then
- \(\int_S^TfdB_t=\int_S^UfdB_t+\int_U^TfdB_t\) almost surely,
- \(\int_S^T(cf+g)\,dB_t=c\int_S^TfdB_t+\int_S^TgdB_t\) almost surely, for constant \(c\),
- \(\mathbb{E}\left[\int_S^TfdB_t\right]=0\),
- \(\int_S^TfdB_t\) is \(\mathcal{F}_T\)-measurable.
These hold for elementary functions by direct computation and extend by \(L^2\)-limits. The integral process \(t\mapsto\int_0^tf\,dB\) is moreover a martingale, and its differential rules are given by the Itô Formula.
7 References
- Øksendal, B. (2003). Stochastic Differential Equations: An Introduction with Applications, 6th ed. Springer, Chapter 3.