## Abstarct

The aim of this work is to provide a unified treatment of moments and cumulants associated with non-linear functionals of completely random measures. A “completely random measure” (also called an “independently scattered random measure”) is a measure ϕ with values in a space of random variables, such that ϕ(
and
where

*A*) and ϕ(*B*) are independent random variables, whenever*A*and*B*are disjoint sets. Examples are Gaussian, Poisson or Gamma random measures. We will specifically focus on multiple stochastic integrals with respect to the random measure ϕ. These integrals are of the form$$
\int\limits_\sigma {f(z_1 ,...,z_n )\phi (dz_1 )...\phi (dz_n )} $$

(1.1.1)

$$
\int\limits_{ \ge \sigma } {f(z_1 ,...,z_n )\phi (dz_1 )...\phi (dz_n )} ,
$$

(1.1.2)

*f*is a symmetric function and ϕ is a completely random measure (for instance, Poisson or Gaussian) on the real line. The integration is not over all of ℝ^{ n }, but over a “diagonal” subset of ℝ^{ n }defined by a partition σ of the integers, …,*n*as illustrated below.## Keywords

Random Measure Stochastic Integration Free Probability Malliavin Calculus Poisson Measure
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag Italia 2011