Moments of Poisson stochastic integrals with random integrands

  1. Nicolas Privault

Abstract

We show that the moment of order n of the Poisson stochastic integral of a random process (ux)x∈X over a metric space X is given by the non-linear Mecke identity
E[( ∫Xux(ω)ω(dx))n]= Σ{P1,...,Pk}∈PnE[ ∫Xkε+sk (u|P1|s1... u|Pk|sk )δ(ds1)...δ(dsk)],
where the sum runs over all partitions P1 ∪ ... ∪ Pk of {1,..., n}, |Pi| denotes the cardinality of Pi, and ε+sk is the operator that acts by addition of points at s1,..., sk to Poisson configurations. This formula recovers known results in case (u(x))x∈X is a deterministic function on X.

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Probability and Mathematical Statistics

32, z. 2, 2012

Pages from 227 to 239

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