**Stochastic Calculus in ***Mathematica*
**Michael Kelly**
University of Western Sydney
Stochastic calculus is an extension of the standard calculus found
in most math
textbooks. But it relies on the development of measure theory as
applied to integration by
Lebesgue. If this measure is the usual probability measure as defined
by Kolmogorov, then
we have a new and very general type of integral, called the Itô
or Stratonovich integral,
which is capable of describing random processes.
The problem with the usual integration is that it can describe only
deterministic
processes. Unfortunately most real-world events are not so easily
represented. In finance,
for instance, the market appears to obey only stochastic or
probabilistic phenomena, so
that only stochastic calculus is an appropriate instrument for its
analysis.
This talk is concerned with presenting a new object-oriented package
for the evaluation of
stochastic differentiation and integration in *Mathematica,* with
a view to applying
the results to financial analysis.
We will describe the definition of stochastic objects with components
relating to their
infinitesimal behavior, giving the name of the process, the underlying
Wiener or driving
process, the drift and dispersion rates, initial value, and time
variable. The main
transformation for such variables is given by Itô's formula, and
the idea is to use this
to describe all possible stochastic variables by applying it directly
to the object's
components and compute their new Itô-transformed
components.
We will present the simulation of stochastic objects by using the
`Random` command from *Mathematica.*
Stochastic integration is
developed so that repeated substitutions of the Itô integral can be
expanded out
to give a Stochastic Taylor
Series representation of any stochastic process in the manner
described by Platen and
Kloeden in their Springer-Verlag texts.
Expected values and stochastic differentiation are possible by
utilizing the underlying
stochastic-object structure to identify differential solutions. The
expected values can be
linked to the `ExpectedValue` definition given in the
Statistics
package.
Mathematical typesetting and formatting for multidimensional Itô
integrals is
accomplished by using Jason Harris's Notation file.
The ultimate purpose of this package is to provide a basis for
stochastic analysis in
finance and a wide range of other science topics such as control
theory, biology, chaos
theory, and economics.
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