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
Here, is the stochastic process describing the state of the system at each time , describes the behavior in the absence of any random shocks and is the sensitivity to the the random noise . The random variables are assumed to be independent and identically distributed with mean zero for each , and defined on an underlying probability space .
Then, should be continuous in and, for each , the sequence of increments should be independent and identically distributed random variables. A possible candidate for 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 cannot exist. We therefore re-express (1) in terms of by integrating it
is easy. Here, is an increasing sequence of times and are random variables. Then, the integral of with respect to is
We could attempt to define the integral with respect to on the right hand side of (2) by approximating 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 be
then the integrals
will tend to infinity as , despite the fact that are all bounded by one. This means that even if we approximate 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 should only depend on the values of for . So, define to be the collection of events in the probability space observable up until time . This is the smallest -algebra with respect to which is measurable for ,
Then, is a filtration of -algebras (http://planetmath.org/FiltrationOfSigmaAlgebras) to which is adapted. If we assume that the piecewise constant process is adapted, then the variables in (1) will be -measurable. Such processes are called elementary predictable, and the following identity is obtained
This is known as the Itô isometry. It ensures that, for bounded and adapted integrands , the integral with respect to cannot become too large, on average. Furthermore, the right hand side is defined for all jointly measurable processes , and allows us extend the integral with respect to 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 -norm, where is the Lebesgue measure on . Equation (5) shows that the map taking the adapted and piecewise constant function to is an isometry with respect to the norms and the -norm . Furthermore, it can be shown, by using the functional monotone class theorem, that every predictable process with finite norm can be approximated in this norm by a sequence elementary predictable processes . That is, the elementary predictable processes are dense (http://planetmath.org/Dense) under the norm, and the Itô integral is defined to be the unique continuous extension (http://planetmath.org/BoundedLinearExtension) from the elementary predictable processes. So, if then,
where convergence is in the 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 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 and any given initial value . When then it is known that does not have finite expectation for and, therefore, does not have finite norm.
The Itô integral can be extended to all predictable processes such that
with probability one, for each . Such processes are said to be -integrable. Given any such , the functional monotone class theorem can be used to show that there exists a sequence of elementary predictable processes such that
in probability (http://planetmath.org/ConvergenceInProbability) for every . 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 additivity, this only simultaneously defines the values of the integral at different times on countable subsets of . The additional constraint is added that the sample paths 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 . 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.
- 1 Bernt Øksendal. , An Introduction with Applications. 5th ed. Springer, 1998.
|Date of creation||2013-03-22 16:13:29|
|Last modified on||2013-03-22 16:13:29|
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