On the combinatorics of iterated stochastic integrals

Abstract

This paper derives several identities for the iterated integrals of a general semimartingale. They involve powers, brackets, exponential and the stochastic exponential. Their form and derivations are combinatorial. The formulae simplify for continuous or finite-variation semimartingales, especially for counting processes. The results are motivated by chaotic representation of martingales, and a simple such application is given.Semimartingale; iterated integrals; power jump processes; Ito's formula; stochastic exponential; chaotic representation