On Itô stochastic integral with respect to vector stable random measures
Fix \(q\in [1,2]\) and a \(q\)-smoothable Banach space \(X\), that is to say satisfying: \(\mathbb{E} [|M_n|^q] \leq C\sum^n_{j=1} \mathbb{E} [|M_j -M_{j-1} |^q]\) for any \(X\)-valued martingale \(M\) and any \(n\in \mathbb{N}\). Fix then \(p<q\), and an \(X\)-valued \(p\)-stable random measure \(Z_p\), with characteristic measure \(Q_p\). The author shows that it exists a stochastic integral on \(L^p(|Q_p |\otimes \mathbb{P})\)-adapted real processes \(u\) on \([0, 1]\): \(u\mapsto \int^1_0 u dZ_p\), extending the obvious integral on simple processes \(u\), and continuous from \(L^p(|Q_p |\otimes \mathbb{P})\) into \(L^0_X (\Omega)\). It is also continuous from \(L^2(|Q_2 |\otimes \mathbb{P})\) into \(L^2_X (\Omega)\), for \(p=q=2\). In this last case, the covariance operator of \(\int^1_0udZ_2\) is \(\int^1_0 \mathbb{E} (|u_t |^2)dQ_2\).
scientific article; zbMATH DE number 1083728