Maximal inequality of Stochastic convolution driven by compensated Poisson random measures in Banach spaces

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Let (E, ·) be a Banach space such that, for some q≥ 2, the function x xq 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 align* u(t)=∫₀ᵗ∫ZS(t-s)ξ(s,z)N(d s,d z), t≥ 0, align* where N is a compensated Poisson random measure on a measurable space (Z,Z) and ξ:[0,∞)×Ω× Z→ E is an F⊗ Z-predictable function. We prove that there exists a càdlàg modification a u of the process u which satisfies the following maximal inequality align* E ₀≤ ₛ≤ ₜ u(s)q′≤ CE (∫₀ᵗ∫Z ξ(s,z) ᵖN(d s,d z))q′/ᵖ, align* for all q′ ≥ q and 1<p≤ 2 with C=C(q,p).