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Stochastic Integrals and Their Expectations
Wilfrid S. Kendall

Expectation

Itô's lemma may be employed in computation of expectations of semimartingale expressions as follows. If we wish to evaluate , then we may expand using Itô's lemma. For well-behaved f (e.g., polynomial functions of Brownian motion), we may replace the differential in the expectation of the stochastic integral by its drift to obtain . If the drift is zero (as is the case for Brownian motion), the integral then vanishes; if the drift is deterministic (e.g., the time process t), then we may simplify further to obtain . It follows by induction that we can evaluate an expectation completely if the semimartingale expression X is a combination of linear operations, multiplication, and stochastic integration performed on time t and n independent Brownian motions (what we called a polynomial semimartingale in the previous section).

ItoIntegralRules therefore defines an expectation operator DoubleStruckCapitalE, which possesses basic linearity properties, and an associated function DoubleStruckCapitalEDoubleStruckCapitalE, which applies transformations of the previous form whenever the semimartingale expression is a polynomial semimartingale.

Examples

We first consider some simple examples of expectations of polynomial semimartingales. Here is a computation of .

Higher-order powers can be dealt with in an equally direct manner, though with increasing computational effort. Here we tabulate for values of n up to 5.

Iterated integrals can be disposed of in a similar fashion. Here we evaluate . Note that DoubleStruckCapitalEDoubleStruckCapitalE can deal with nonpolynomial functions of t.

Here we evaluate .

Nonpolynomial semimartingales are left unsimplified if the nonpolynomial part involves Brownian motions, as in this evaluation of .

There are further techniques available for dealing with nonpolynomial semimartingales. For example, consider the evaluation of . We could of course evaluate this directly using the density for the random variable B, and this itself can be automated using mathStatica [11]. However, we can also make progress using two further ItoIntegralRules (ItoExpectItoIntegralRule, ItoExpectExpandRule), which are components of DoubleStruckCapitalEDoubleStruckCapitalE. The first of these implements the interplay between expectation and drift described at the start of this section, while the second expands DoubleStruckCapitalE linearly and extracts nonrandom terms.

We have thus obtained a recursive expression for that can now be used to form a differential equation by further developing this code.

So we obtain

Further examples can be found in ItoIntegralTests.nb. This differential equation approach can be applied even when we require the expectation of a quantity that is not a simple function of Brownian motion. See, for example, the treatment of the distribution of the stochastic area integral in [2].

Calculations for the Ornstein-Uhlenbeck Process

The original motivation for this work was to provide an environment to aid computations of expectations of quantities associated with the integrated Ornstein-Uhlenbeck process. To illustrate this, we use a pair of stochastic differential equations

to define an Ornstein-Uhlenbeck process and its integrated variant. To do this in Itovsn3 we use Itosde.

Note that it must be stated explicitly that both Alpha and the initial values , are Constant in time. It is possible to solve these linear stochastic differential equations in closed form: and . We express the solutions using ItoIntegral:

and

and verify directly that they satisfy the relevant stochastic differential equations:

We now compute mean values

and the variance-covariance matrix (using ).

Computation of the fourth central moment is equally direct.

The image analysis application required the derivation of the conditional distribution of X and U at a specified time s given the values of X and U at 0 and t, for . Since is a Gaussian process, we can find this by straightforward use of the Statistics`MultinormalDistribution` package once the means and variance-covariance matrix are calculated for the various values of X and U at times 0, s, and t. Of course it is possible to derive the conditional distribution by hand; an advantage of working in Mathematica is that we are then able to proceed directly to simulation experiments and so forth.



     
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