Categories: Mathematics, Stochastic analysis.

Itō integral

The Itō integral offers a way to integrate a given stochastic process \(G_t\) with respect to a Wiener process \(B_t\), which is also a stochastic process. The Itō integral \(I_t\) of \(G_t\) is defined as follows:

\[\begin{aligned} \boxed{ I_t \equiv \int_a^b G_t \dd{B_t} \equiv \lim_{h \to 0} \sum_{t = a}^{t = b} G_t \big(B_{t + h} - B_t\big) } \end{aligned}\]

Where have partitioned the time interval \([a, b]\) into steps of size \(h\). The above integral exists if \(G_t\) and \(B_t\) are adapted to a common filtration \(\mathcal{F}_t\), and \(\mathbf{E}[G_t^2]\) is integrable for \(t \in [a, b]\). If \(I_t\) exists, \(G_t\) is said to be Itō-integrable with respect to \(B_t\).

Motivation

Consider the following simple first-order differential equation for \(X_t\), for some function \(f\):

\[\begin{aligned} \dv{X_t}{t} = f(X_t) \end{aligned}\]

This can be solved numerically using the explicit Euler scheme by discretizing it with step size \(h\), which can be applied recursively, leading to:

\[\begin{aligned} X_{t+h} \approx X_{t} + f(X_t) \: h \quad \implies \quad X_t \approx X_0 + \sum_{s = 0}^{s = t} f(X_s) \: h \end{aligned}\]

In the limit \(h \to 0\), this leads to the following unsurprising integral for \(X_t\):

\[\begin{aligned} \int_0^t f(X_s) \dd{s} = \lim_{h \to 0} \sum_{s = 0}^{s = t} f(X_s) \: h \end{aligned}\]

In contrast, consider the stochastic differential equation below, where \(\xi_t\) represents white noise, which is informally the \(t\)-derivative of the Wiener process \(\xi_t = \dv*{B_t}{t}\):

\[\begin{aligned} \dv{X_t}{t} = g(X_t) \: \xi_t \end{aligned}\]

Now \(X_t\) is not deterministic, since \(\xi_t\) is derived from a random variable \(B_t\). If \(g = 1\), we expect \(X_t = X_0 + B_t\). With this in mind, we introduce the Euler-Maruyama scheme:

\[\begin{aligned} X_{t+h} &= X_t + g(X_t) \: (\xi_{t+h} - \xi_t) \: h \\ &= X_t + g(X_t) \: (B_{t+h} - B_t) \end{aligned}\]

We would like to turn this into an integral for \(X_t\), as we did above. Therefore, we state:

\[\begin{aligned} X_t = X_0 + \int_0^t g(X_s) \dd{B_s} \end{aligned}\]

This integral is defined as below, analogously to the first, but with \(h\) replaced by the increment \(B_{t+h} \!-\! B_t\) of a Wiener process. This is an Itō integral:

\[\begin{aligned} \int_0^t g(X_s) \dd{B_s} \equiv \lim_{h \to 0} \sum_{s = 0}^{s = t} g(X_s) \big(B_{s + h} - B_s\big) \end{aligned}\]

For more information about applying the Itō integral in this way, see the Itō calculus.

Properties

Since \(G_t\) and \(B_t\) must be known (i.e. \(\mathcal{F}_t\)-adapted) in order to evaluate the Itō integral \(I_t\) at any given \(t\), it logically follows that \(I_t\) is also \(\mathcal{F}_t\)-adapted.

Because the Itō integral is defined as the limit of a sum of linear terms, it inherits this linearity. Consider two Itō-integrable processes \(G_t\) and \(H_t\), and two constants \(v, w \in \mathbb{R}\):

\[\begin{aligned} \int_a^b v G_t + w H_t \dd{B_t} = v\! \int_a^b G_t \dd{B_t} +\: w\! \int_a^b H_t \dd{B_t} \end{aligned}\]

By adding multiple summations, the Itō integral clearly satisfies, for \(a < b < c\):

\[\begin{aligned} \int_a^c G_t \dd{B_t} = \int_a^b G_t \dd{B_t} + \int_b^c G_t \dd{B_t} \end{aligned}\]

A more interesting property is the Itō isometry, which expresses the expectation of the square of an Itō integral of \(G_t\) as a simpler “ordinary” integral of the expectation of \(G_t^2\) (which exists by the definition of Itō-integrability):

\[\begin{aligned} \boxed{ \mathbf{E} \bigg( \int_a^b G_t \dd{B_t} \bigg)^2 = \int_a^b \mathbf{E} \big[ G_t^2 \big] \dd{t} } \end{aligned}\]

Furthermore, Itō integrals are martingales, meaning that the average noise contribution is zero, which makes intuitive sense, since true white noise cannot be biased.

References

  1. U.H. Thygesen, Lecture notes on diffusions and stochastic differential equations, 2021, Polyteknisk Kompendie.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.
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