Probability and Mathematical Statistics: Vol. 3, Fasc. 2

ON MULTIPLE POISSON STOCHASTIC INTEGRALS AND ASSOCIATED MARKOV SEMIGROUPS

D. Surgailis

Abstract: Multiple stochastic integrals (m.s.i.)

(n) integral q (f) = X f(x1,...,xn)q(dx1)...q(dxn), n = l,2,... n

with respect to the centered Poisson random measure $q(dx),$ $E[q(dx)] = 0,$ $E[(q(dx))] = m(dx),$ are discussed, where $(X, m)$ is a measurable space. A ”diagram formula” for evaluation of products of (Poisson) m.s.i. as sums of m.s.i. is derived. With a given contraction semigroup $At,$ $t > 0,$ in $2 L (X)$ we associate a semigroup $G(At),$ $t > 0,$ in $2 L (_O_)$ by the relation

(n) (n) G(At)q (f1^ ox ... ox ^fn) = q (Atf1^ ox ...^ ox Atfn)

and prove that $G(At),$ $t > 0,$ is Markov if and only if $At,$ $t > 0,$ is doubly sub-Markov; the corresponding Markov process can be described as time evolution (with immigration) of the (infinite) system of particles, each moving independently according to $At,$ $t > 0.$

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -