In order to introduce a stochastic process as a solution of a stochastic differential equation as we introduce the concept of the Itô-integral: a stochastic integral with respect to a Wiener process. Formally the construction of the Itô-integral is similar to the one of the Stieltjes-integral. But instead of integrating with respect to a deterministic function (Stieltjes-integral), the Itô-integral integrates with respect to a random function, more precisely, the path of a Wiener process. Since the integrant itself can be random, i.e. it can be a path of a stochastic process, one has to analyze the mutual dependencies of the integrant and the Wiener process.
Let
be the process to integrate and let
be a standard Wiener process. The
definition of a stochastic integral assumes that
is not anticipating. Intuitively, it means
that the process up to time
does not contain any information
about future increments
of the
Wiener process. In particular,
is independent of
An integral of a function is usually defined as the limit of the
sum of the suitably weighted function. Similarly, the Itô
integral with respect to a Wiener
process is defined as the limit of the sum of the (randomly)
weighted (random) function
It is not hard to be more precise on the non anticipating property
of
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(information grows as time |
evolves) | ||
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about events occurring after | ||
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(the Wiener process is adapted to |
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The integral depends crucially on the point of the interval
at which the random variable
is evaluated in (5.1). Consider the example
i.e. we integrate the Wiener process with
respect to itself. As a gedankenexperiment we replace in
(5.1)
by an arbitrary point
of
the interval
If we
defined:
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To illustrate how Itô-integrals are computed, and that other than the usual computation rules have to be applied, we show that:
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