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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.