Itô integral

1 The need for a stochastic integral

Suppose that we wish to model a system which is continuously subject to random shocks. Then, we may want to consider a differential equation of the form

$$\frac{d{X}_t}{dt}=b(X_t)+a(X_t){\xi }_t.$$ (1)

Here, $X_t$ is the stochastic process describing the state of the system at each time $t\ge 0$, $b$ describes the behavior in the absence of any random shocks and $a$ is the sensitivity to the the random noise ${\xi }_t$. The random variables ${\xi }_t$ are assumed to be independent and identically distributed with mean zero for each $t\ge 0$, and defined on an underlying probability space $(\mathrm{\Omega },ℱ,\mathbb{P})$.

The first problem is that it is not clear how to define $\xi$ continuously in time such that (1) can be solved. Let $W$ be the integral

$$W_t={\int }_0^t{\xi }_s𝑑s.$$

Then, $W_t$ should be continuous in $t$ and, for each $h>0$, the sequence of increments ${(W_{(n+1)h}-W_{nh})}_{n=0,1,\mathrm{\dots }}$ should be independent and identically distributed random variables. A possible candidate for $W$ is Brownian motion, and it can be shown that, up to a constant scaling factor, this is the only possibility. However, Brownian motion is nowhere differentiable and ${\xi }_t=d{W}_t/dt$ cannot exist. We therefore re-express (1) in terms of $W$ by integrating it

$$X_t=X_0+{\int }_0^{t}b(X_s)𝑑s+{\int }_0^{t}a(X_s)𝑑W_s.$$ (2)

The idea of Itô integration is to give meaning to the final integral on the right hand side of (2). Defining the integral with respect to piecewise constant functions of the form

(3)

is easy. Here, $0\le t_0\le t_1\le \mathrm{\cdots }\le t_n$ is an increasing sequence of times and $c_k$ are random variables. Then, the integral of $\alpha$ with respect to $W$ is

$${\int }_0^t{\alpha }_s𝑑W_s=\sum _{k=1}^n{c}_k\left(W_{t_k\wedge t}-W_{t_{k-1}\wedge t}\right).$$ (4)

We could attempt to define the integral with respect to $W$ on the right hand side of (2) by approximating $a(X_t)$ by such piecewise constant functions. However, the sample paths of Brownian motion are of infinite total variation over all intervals. This means that if we let ${\alpha }^n$ be

then the integrals

$${\int }_0^1{\alpha }^n𝑑W=\sum _{k=1}^n\left|W_{k/n}-W_{(k-1)/n}\right|$$

will tend to infinity as $n\to \mathrm{\infty }$, despite the fact that ${\alpha }^n$ are all bounded by one. This means that even if we approximate $a(X_t)$ as closely as we like by piecewise constant functions, the integral can diverge to infinity.

2 The Itô integral

Itô’s solution was to first note that if we can define the integral in a way such that (2) has a unique solution, then $X_t$ should only depend on the values of $W_s$ for $s\le t$. So, define ${ℱ}_t$ to be the collection of events in the probability space observable up until time $t$. This is the smallest $\sigma$-algebra with respect to which $W_s$ is measurable for $s\le t$,

$${ℱ}_t=\sigma (W_s:s\le t).$$

Then, ${({ℱ}_t)}_{t\in {ℝ}_{+}}$ is a filtration of $\sigma$-algebras (http://planetmath.org/FiltrationOfSigmaAlgebras) to which $W$ is adapted. If we assume that the piecewise constant process ${\alpha }_t$ is adapted, then the variables $c_k$ in (1) will be ${ℱ}_{t_{k-1}}$-measurable. Such processes are called elementary predictable, and the following identity is obtained

$$𝔼\left[{\left({\int }_0^t\alpha 𝑑W\right)}^2\right]=𝔼\left[{\int }_0^t{\alpha }_s^2𝑑s\right].$$ (5)

This is known as the Itô isometry. It ensures that, for bounded and adapted integrands $\alpha$, the integral with respect to $W$ cannot become too large, on average. Furthermore, the right hand side is defined for all jointly measurable processes $\alpha$, and allows us extend the integral with respect to $W$ to integrands which can be approximated by piecewise constant and adapted processes. Such processes are called predictable and include all continuous and adapted processes, so it gives meaning to (2). If we define

$${\parallel \alpha \parallel }_{2,t}\equiv 𝔼{\left[{\int }_0^t{\alpha }_s^2𝑑s\right]}^{\frac{1}{2}}$$

then this is a norm on the set of predictable processes — in fact, it is the $L^2(\mathbb{P}\times \lambda )$-norm, where $\lambda$ is the Lebesgue measure on $[0,t]$. Equation (5) shows that the map taking the adapted and piecewise constant function $\alpha$ to ${\int }_0^t\alpha 𝑑W$ is an isometry with respect to the norms $\parallel \cdot {\parallel }_{2,t}$ and the $L^2(\mathbb{P})$-norm $\parallel \cdot {\parallel }_2$. Furthermore, it can be shown, by using the functional monotone class theorem, that every predictable process $\alpha$ with finite $\parallel \cdot {\parallel }_{2,t}$ norm can be approximated in this norm by a sequence elementary predictable processes ${\alpha }^n$. That is, the elementary predictable processes are dense (http://planetmath.org/Dense) under the $\parallel \cdot {\parallel }_{2,t}$ norm, and the Itô integral is defined to be the unique continuous extension (http://planetmath.org/BoundedLinearExtension) from the elementary predictable processes. So, if ${\parallel \alpha -{\alpha }^n\parallel }_{2,t}\to 0$ then,

$${\int }_0^t\alpha 𝑑W=\underset{n\to \mathrm{\infty }}{lim}{\int }_0^t{\alpha }^n𝑑W$$

where convergence is in the $L^2(\mathbb{P})$ norm.

3 Extension of the integral

The definition above gives a construction of the Itô integral which is useful in many applications. However, the restriction that ${\parallel \alpha \parallel }_{2,t}$ must be finite is too restrictive for some situations. For example, it is possible to construct solutions to the stochastic differential equation

$$dX=X^{c}dW$$

for any constant $c>0$ and any given initial value $X_0>0$. When $c>1$ then it is known that $X_t^{2c}$ does not have finite expectation for $t>0$ and, therefore, $X^c$ does not have finite $\parallel \cdot {\parallel }_{2,t}$ norm.

The Itô integral can be extended to all predictable processes $\alpha$ such that

with probability one, for each $t>0$. Such processes are said to be $W$-integrable. Given any such $\alpha$, the functional monotone class theorem can be used to show that there exists a sequence of elementary predictable processes ${\alpha }^n$ such that

$${\int }_0^t{({\alpha }_s^n-{\alpha }_s)}^2𝑑s\to 0$$

in probability (http://planetmath.org/ConvergenceInProbability) for every $t>0$. Then, using the Itô isometry it can be shown that the limit

$${\int }_0^t\alpha 𝑑W\equiv \underset{n\to \mathrm{\infty }}{lim}{\int }_0^t{\alpha }^n𝑑W$$

exists, where convergence is in probability. Finally, we note that that this only defines the integral $\mathbb{P}$-almost everywhere, at each time. However, as probability measures only satisfy countable additivity, this only simultaneously defines the values of the integral at different times $t$ on countable subsets of ${ℝ}_{+}$. The additional constraint is added that the sample paths $t\mapsto {\int }_0^t\alpha 𝑑W$ are continuous, and it can be shown that it is possible to take such continuous versions of the integral. This defines the integral simultaneously at all times up to a zero probability set.

We have presented a definition of the Itô integral with respect to a Brownian motion $W$. Then, an important result for manipulating such integrals is Itô’s lemma (http://planetmath.org/ItosFormula). Furthermore, it is sometimes necessary to define integration for more general processes. See stochastic integral (http://planetmath.org/StochasticIntegration) for a definition of the integral with respect to general semimartingales, which includes discontinuous processes, such as Lévy processes.

Title	Itô integral
Canonical name	ItoIntegral
Date of creation	2013-03-22 16:13:29
Last modified on	2013-03-22 16:13:29
Owner	stevecheng (10074)
Last modified by	stevecheng (10074)
Numerical id	21
Author	stevecheng (10074)
Entry type	Topic
Classification	msc 60H10
Synonym	Ito integral
Synonym	stochastic integral
Related topic	StochasticCalculusAndSDE
Related topic	StochasticIntegration
Defines	Itô integral