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}

## 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.