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GR/K71677 SUMMARY: Stochastic calculus in AXIOM using modules of stochastic differentials

Wilfrid S Kendall

Department of Statistics, University of Warwick, Coventry CV4 7AL, UK
An implementation of stochastic calculus has been made within AXIOM, taking advantage of its innovative features including information-hiding and programming in terms of mathematical structures. For example stochastic differentials are defined in AXIOM as a domain of computation which is a rng (ring without unity) and module over the algebra of bounded predictable functions. The advantages of the AXIOM approach include reliability, because one is programming with structures corresponding directly to the underlying mathematical theory, and flexibility. The implementation, together with supporting documentation, has been made available on the WorldWide Web at

(where also can be found associated preprints). The implementation is now being used by the investigator to attack problems relating to stochastic perpetuities in mathematical finance. Other work facilitated by the project has contributed to the statistical theory of shape, and has been influential in probability theory at the foundations of perfect simulation and in stochastic analysis. Finally, in very recent work, the AXIOM implementation of stochastic calculus has motivated work on algebraic perspectives on stochastic calculus which is likely to feed back into algorithmic developments for symbolic Itô calculus: it has been shown that suitable classes of semimartingales can be used to generate algebraically closed fields of (generalized) semimartingales, opening up a promise of stochastic differential algebra which is now being vigorously pursued.

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