Mathematica Journal
Volume 9, Issue 4


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Stochastic Integrals and Their Expectations
Wilfrid S. Kendall


We have shown in this article (and in ItoIntegralTests.nb) how the package Itovsn3 can be extended to provide the ability to manipulate stochastic integrals and compute expectations of stochastic calculus quantities. Further extensions are possible--we have already noted Gaines' work [9, 10] on simplification of polynomial semimartingales based on several independent Brownian motions. A more ambitious extension would be to generalize and automate the differential equation strategy described here for computing expectations, such as . This would be a demanding project involving the automatic recognition of differential equations satisfied by the expectation in question. We have also noted the possibility of interaction with other packages such as mathStatica. For example, Colin Rose has pointed out how the definition of DoubleStruckCapitalEDoubleStruckCapitalE can be modified so as to invoke mathStatica's Expect function when the expression has been determined not to be a polynomial semimartingale. This would provide a direct means of computing ; however, differential equation techniques would still be required when dealing with expressions involving stochastic integrals.

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