2010/05/10 by Jiahui Zhu, Zdzisław Brzeźniak, Zhu, Jiahui +3
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60F10 #60G57 #60H05 #60H15 #60J75 #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #math.PR #msc:60F10 #msc:60G57 #msc:60H05 #msc:60H15 #msc:60J75
paper · pdf · doi:10.48550/arxiv.1005.1600
This version is only very slightly updated as compared to the one from September 2015
openalex publication_date 2010/05/10 · arxiv created 2015/10/22 · arxiv updated 2015/10/23 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Let (E, ‖ ⋅‖) be a Banach space such that, for some q≥ 2, the function x↦ ‖x‖q is of C2 class and its first and second Fréchet derivatives are bounded by some constant multiples of (q-1)-th power of the norm and (q-2)-th power of the norm and let S be a C0-semigroup of contraction type on (E, ‖ ⋅‖). We consider the following stochastic convolution process u(t)=∫0t∫ZS(t-s)ξ(s,z) N(d s,d z), t≥ 0, where N is a compensated Poisson random measure on a measurable space (Z,Z) and ξ:[0,∞)×Ω× Z→ E is an \mathbbF⊗ Z-predictable function. We prove that there exists a càdlàg modification a u of the process u which satisfies the following maximal inequality 𝔼 sup0≤ s≤ t ‖u(s)‖q^′≤ C 𝔼 (∫0t∫Z ‖ξ(s,z) ‖p N(d s,d z))(q^′)/(p), for all q^′ ≥ q and 1<p≤ 2 with C=C(q,p).