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July 1997 Stochastic integrals: a combinatorial approach
Gian-Carlo Rota, Timothy C. Wallstrom
Ann. Probab. 25(3): 1257-1283 (July 1997). DOI: 10.1214/aop/1024404513


A combinatorial definition of multiple stochastic integrals is given in the setting of random measures. It is shown that some properties of such stochastic integrals, formerly known to hold in special cases, are instances of combinatorial identities on the lattice of partitions of a set. The notion of stochastic sequences of binomial type is introduced as a generalization of special polynomial sequences occuring in stochastic integration, such as Hermite, Poisson–Charlier and Kravchuk polynomials. It is shown that identities for such polynomial sets have a common origin.


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Gian-Carlo Rota. Timothy C. Wallstrom. "Stochastic integrals: a combinatorial approach." Ann. Probab. 25 (3) 1257 - 1283, July 1997.


Published: July 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0886.60046
MathSciNet: MR1457619
Digital Object Identifier: 10.1214/aop/1024404513

Primary: 05A18 , 60H05
Secondary: 05E05 , 05E35 , 11B65 , 60G57 , 81T18

Keywords: discrete and homogeneous chaos , Kailath-Segall formula , multiple stochastic integrals , orthogonal polynomials , partitions of sets , symmetric functions

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 3 • July 1997
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