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 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. 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.

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.

Unfortunately most real-world events are not so easily represented. The problem with the usual integration is that it can describe only deterministic processes. 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. But it relies on the development of measure theory as applied to integration by Lebesgue. Stochastic calculus is an extension of the standard calculus found in most math textbooks.

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Stochastic calculus with infinitesimals