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The algebra of iterated stochastic integrals. (English) Zbl 0827.60038
Summary: Using results from work on shuffle algebras, we show that Lyndon words provide an algebraic basis for the sets of iterated Stratonovich or ItĂ´ 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 H. J. Sussmann [in: Stochastic differential systems, stochastic control theory and applications. IMA Vol. Math. Appl. 10, 563-582 (1988; Zbl 0651.60065)] 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 P. E. Kloeden and E. Platen [“Numerical solutions of stochastic differential equations” (1992; Zbl 0752.60043)] using a purely combinatorial approach.

MSC:
60H05 Stochastic integrals
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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