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 ∈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),t∈T where b is a deterministic drift function and $$X(t) = \int_S f(t,s) \left[N(ds)-(|f(t,s)|\vee 1)^{-1}\nu(ds)\right].$$ Here N is a Poisson random measure on a Borel space S with σ−finite mean measure ν, and $$f{:} T \times S \mapsto R$$ is a measurable deterministic function. Let τ: T2 → R+ be a continuous pseudo–metric on T. Define the τ-Lipschitz norm of the sections of f by $$ \|f\|_{\tau} (s)=D^{-1}f(t_0,s) + \sup\limits_{u,v\in T} {{|f(u,s)-f(v,s)|}\over{\tau(u,v)}}$$ for some t0 ∈T, where D is the diameter of (T,τ). The sufficient conditions for boundedness and continuity of X are given in terms of the measure $$\nu,\||f\||_\tau$$ and majorizing measure and or metric entropy conditions determined by τ. They are applied to stochastic integrals of the form $$Y(t) = \int_S g(t,s)M(ds)\quad t\in T,$$ 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 $$\sup_{t\in T}|X(t)|$$ and $$\sup_{\tau (t,u)\leqslant\delta,t,u \in T}|X(t) - X(u)|$$ for all $$0