Continuity and boundedness of infinitely divisible processes: A Poisson point process approach

Sufficient conditions for boundedness and continuity are obtained for stochastically continuous infinitely divisible processes, without Gaussian component, {Y(t),t is an element of T}, where T is a compact metric space or pseudo-metric space. Such processes have a version given by Y(t)=X(t)+b(t),is an element of T where b is a deterministic drift function and X(t) = integral(S) f(t,s) [N(ds) - (vertical bar f(t,s)vertical bar boolean OR 1)(-1)nu(ds)]. Here N is a Poisson random measure on a Borel space S with sigma-finite mean measure nu, and f : T x S (bar right arrow) R is a measurable deterministic function. Let tau: T-2 -> R+ be a continuous pseudo-metric on T. Define the tau-Lipschitz norm of the sections of f by [GRAPHICS] for some t(0) is an element of T, where D is the diameter of (T, tau). The sufficient conditions for boundedness and continuity of X are given in terms of the measure nu, parallel to f parallel to(tau) and majorizing measure and or metric entropy conditions determined by tau. They are applied to stochastic integrals of the form Y(t) = integral(S) g(t,s)M(ds) t is an element of T, where M is a zero-mean, independently scattered, infinitely divisible random measure without Gaussian component. Several examples are given which show that in many cases the conditions obtained are quite sharp. In addition to obtaining conditions for continuity and boundedness, bounds are obtained for the weak and strong L-p norms of sup(t is an element of T) vertical bar X( t)vertical bar and sup(tau(t,u)) (<=) (delta,t,u is an element of T) vertical bar X(t) - X(u)vertical bar for all 0 < delta <= D. These results depend on inequalities for moments and related functions of the weak and strong l(p) norms of sequences {x(j)}, which are the events of Poisson point process M on R+ and are given in terms of the intensity measure of M. These results are of independent interest.