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 equationMathworldPlanetmath of the form

dXtdt=b(Xt)+a(Xt)ξt. (1)

Here, Xt is the stochastic processMathworldPlanetmath describing the state of the system at each time t0, b describes the behavior in the absence of any random shocks and a is the sensitivity to the the random noise ξt. The random variablesMathworldPlanetmath ξt are assumed to be independent and identically distributed with mean zero for each t0, and defined on an underlying probability spaceMathworldPlanetmath (Ω,,).

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


Then, Wt should be continuousMathworldPlanetmathPlanetmath in t and, for each h>0, the sequence of increments (W(n+1)h-Wnh)n=0,1, should be independent and identically distributed random variables. A possible candidate for W is Brownian motionMathworldPlanetmath, and it can be shown that, up to a constant scaling factor, this is the only possibility. However, Brownian motion is nowhere differentiableMathworldPlanetmathPlanetmath and ξt=dWt/dt cannot exist. We therefore re-express (1) in terms of W by integrating it

Xt=X0+0tb(Xs)𝑑s+0ta(Xs)𝑑Ws. (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

αt=k=1nck1{tk-1<ttk} (3)

is easy. Here, 0t0t1tn is an increasing sequence of times and ck are random variables. Then, the integral of α with respect to W is

0tαs𝑑Ws=k=1nck(Wtkt-Wtk-1t). (4)

We could attempt to define the integral with respect to W on the right hand side of (2) by approximating a(Xt) by such piecewise constant functions. However, the sample paths of Brownian motion are of infiniteMathworldPlanetmathPlanetmath total variationMathworldPlanetmathPlanetmath over all intervals. This means that if we let αn be


then the integrals


will tend to infinityMathworldPlanetmath as n, despite the fact that αn are all bounded by one. This means that even if we approximate a(Xt) 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 Xt should only depend on the values of Ws for st. So, define t to be the collectionMathworldPlanetmath of events in the probability space observable up until time t. This is the smallest σ-algebra with respect to which Ws is measurable for st,


Then, (t)t+ is a filtrationPlanetmathPlanetmath of σ-algebras ( to which W is adapted. If we assume that the piecewise constant process αt is adapted, then the variables ck in (1) will be tk-1-measurable. Such processes are called elementary predictable, and the following identity is obtained

𝔼[(0tα𝑑W)2]=𝔼[0tαs2𝑑s]. (5)

This is known as the Itô isometry. It ensures that, for bounded and adapted integrands α, the integral with respect to W cannot become too large, on averageMathworldPlanetmath. Furthermore, the right hand side is defined for all jointly measurable processes α, 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


then this is a norm on the set of predictable processes — in fact, it is the L2(×λ)-norm, where λ is the Lebesgue measureMathworldPlanetmath on [0,t]. Equation (5) shows that the map taking the adapted and piecewise constant function α to 0tα𝑑W is an isometry with respect to the norms 2,t and the L2()-norm 2. Furthermore, it can be shown, by using the functional monotone class theorem, that every predictable process α with finite 2,t norm can be approximated in this norm by a sequence elementary predictable processes αn. That is, the elementary predictable processes are dense ( under the 2,t norm, and the Itô integral is defined to be the unique continuous extensionPlanetmathPlanetmath ( from the elementary predictable processes. So, if α-αn2,t0 then,


where convergence is in the L2() norm.

3 Extension of the integral

The definition above gives a construction of the Itô integral which is useful in many applications. However, the restrictionPlanetmathPlanetmath that α2,t must be finite is too restrictive for some situations. For example, it is possible to construct solutions to the stochastic differential equation


for any constant c>0 and any given initial value X0>0. When c>1 then it is known that Xt2c does not have finite expectation for t>0 and, therefore, Xc does not have finite 2,t norm.

The Itô integral can be extended to all predictable processes α such that


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


in probability ( for every t>0. Then, using the Itô isometry it can be shown that the limit


exists, where convergence is in probability. Finally, we note that that this only defines the integral -almost everywhere, at each time. However, as probability measures only satisfy countable additivityMathworldPlanetmath, this only simultaneously defines the values of the integral at different times t on countableMathworldPlanetmath subsets of +. The additional constraint is added that the sample paths t0tα𝑑W 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 ( Furthermore, it is sometimes necessary to define integration for more general processes. See stochastic integral ( for a definition of the integral with respect to general semimartingales, which includes discontinuousMathworldPlanetmath processes, such as Lévy processes.


  • 1 Bernt Øksendal. , An Introduction with Applications. 5th ed. Springer, 1998.
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