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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 32, Fasc. 2,
pages 227 - 239
 

MOMENTS OF POISSON STOCHASTIC INTEGRALS WITH RANDOM INTEGRANDS

Nicolas Privault

Abstract: We show that the moment of order n of the Poisson stochastic integral of a random process (u ) x x∈X over a metric space X is given by the non-linear Mecke identity
∫ E[( ux(ω )ω (dx))n] X ∑ ∫ = E [ ε+𝔰k(u |Ps11|...u|sPkk|)σ(ds1)...σ(dsk)], (P1,...,Pk)∈Pn Xk

where the sum runs over all partitions P1 ∪ ...∪ Pk of (1,...,n) , |Pi| denotes the cardinality of Pi , and + ε𝔰k 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 .

2000 AMS Mathematics Subject Classification: Primary: 60G57; Secondary: 60G55, 60H07.

Keywords and phrases: Poisson stochastic integrals, moment identities, Bell polynomials, Poisson–Skorohod integral.

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