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A basis for iterated stochastic integrals

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  • Gaines, J.G.

Abstract

We explore the algebra of the sets of iterated Stratonovich or Ito integrals that appear in the stochastic Taylor series expansion of the solution to a stochastic differential equation. The algebra of iterated Stratonovich integrals with pointwise multiplication is a shuffle algebra, allowing us to apply results from work on shuffle algebras. The pointwise product of Ito integrals is a modified shuffle product. Lyndon words provide an algebraic basis for both sets of iterated integrals. This basis is similar to, but simpler than, that obtained by Sussmann using Hall words.

Suggested Citation

  • Gaines, J.G., 1995. "A basis for iterated stochastic integrals," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 38(1), pages 7-11.
    • Handle: RePEc:eee:matcom:v:38:y:1995:i:1:p:7-11
    DOI: 10.1016/0378-4754(93)E0061-9
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    References listed on IDEAS

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    1. P. E. Kloeden & Eckhard Platen, 1991. "Stratonovich and Ito Stochastic Taylor Expansions," Published Paper Series 1991-2, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
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    Cited by:

    1. Tocino, A., 2009. "Multiple stochastic integrals with Mathematica," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(5), pages 1658-1667.

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