Jamshidian, Farshid (2008): On the combinatorics of iterated stochastic integrals.

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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 finitevariation semimartingales, especially for counting processes. The results are motivated by chaotic representation of martingales, and a simple such application is given.
Item Type:  MPRA Paper 

Original Title:  On the combinatorics of iterated stochastic integrals 
Language:  English 
Keywords:  Semimartingale; iterated integrals; power jump processes; Ito's formula; stochastic exponential; chaotic representation 
Subjects:  C  Mathematical and Quantitative Methods > C0  General 
Item ID:  7165 
Depositing User:  Farshid Jamshidian 
Date Deposited:  15. Feb 2008 09:00 
Last Modified:  18. Feb 2013 12:48 
References:  [1] Itˆo, K.: Mutliple Wiener Integral. J. Math Soc. Japan 3., 157164, (1951). [2] Jamshidian, F.: Chaotic expansion of powers and martingales representation, working paper (2005). [3] Nualart, D. and Schoutens, W.: Chaotic and predictable representations for Levy processes. Stochastic Processes and their Applications 90, 109122, (2000). [4] Oertel, F.: A Hilbert space approach to Wiener Chaos Decomposition and applications to finance, working paper (2003). [5] Protter, P.: Stochastic integration and dierential equations. Springer, second edition (2005). [6] Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Spriner (1991). [7] Wikipedia, online encyclopedia, http://en.wikipedia.org/wiki/Stirling number. [8] Yan Yip, W., Stephens, D., and Olhede S.: The Explicit Chaotic Representation of the powers of increments of L´evy Processes. Statistics Section, Technical Report TR0704, (2007) 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/7165 