The algebra of iterated stochastic integrals

Abstract

Using results from work on shuffle algebras, we show that Lyndon words provide an algebraic basis for the sets of iterated Stratonovich or Ito integrals that appear in the stochastic Taylor series expansion of the solution to a stochastic differential equation and give a method for rewriting these stochastic integrals in terms of the basis. This basis is similar to, but simpler than, that obtained by Sussmann ([19]) using Hall words. We also show how the shuffle product can be used to obtain moments of stochastic integrals and hence reprove a result of Kloeden and Platen ([9]) using a purely combinatorial approach.