Multiplication formulae

Abstarct

The forthcoming Theorem 6.1.1 applies to every good completely random measure ϕ. It gives a universal combinatorial rule, according to which every product of multiple stochastic integrals can be represented as a sum over diagonal measures that are indexed by non-flat diagrams (as defined in Section 4.1). We will see that product formulae are crucial in order to deduce explicit expressions for the cumulants and the moments of multiple integrals. As discussed later in this chapter, Theorem 6.1.1 contains (as special cases) two celebrated product formulae for integrals with respect to Gaussian and Poisson random measures. We provide two proofs of Theorem 6.1.1: the first one is new and it is based on a decomposition of partially diagonal sets; the second consists in a slight variation of the combinatorial arguments displayed in the proofs of [132, Th. 3 and Th. 4], and is included for the sake of completeness. The theorem is formulated for simple kernels to ensure that the integrals are always defined, in particular the quantity St ^ϕ,[n]_σ which appears on the RHS of (6.1.2).