Hannebicque, Brice; Herbin, Érick Regularity of an abstract Wiener integral. (English) Zbl 1510.60042 Stochastic Processes Appl. 154, 154-196 (2022). Summary: In this article, we propose a way to study sample path properties of processes indexed by a general poset \((\mathcal{T},\preccurlyeq)\). This framework encompasses large classes of vector spaces, manifolds and continuous \(\mathbb{R}\)-trees. We define a Wiener-type integral \(Y_t=\int_{\preccurlyeq t}fdX\) for all \(t\in\mathcal{T}\), a deterministic function \(f:\mathcal{T}\to\mathbb{R}\) and a set-indexed Lévy process \(X\). Bounds for the Hölder regularity of \(Y\) are given which indicate how the regularities of \(f\) and \(X\) contributes to that of \(Y\). This work is a continuation of E. Herbin and A. Richard [Isr. J. Math. 215, No. 1, 397–440 (2016; Zbl 1368.60042)] and extends those of S. Jaffard [Probab. Theory Relat. Fields 114, No. 2, 207–227 (1999; Zbl 0947.60039)] and P. Balança and E. Herbin [Stochastic Processes Appl. 122, No. 6, 2346–2382 (2012; Zbl 1263.60032)]. MSC: 60H05 Stochastic integrals 60G10 Stationary stochastic processes 60G17 Sample path properties 60G20 Generalized stochastic processes 60G51 Processes with independent increments; Lévy processes 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60G57 Random measures Keywords:sample path properties; Hölder regularity; set-indexed process; stochastic integral; random measure; Lévy process; stationarity Citations:Zbl 1368.60042; Zbl 0947.60039; Zbl 1263.60032 PDFBibTeX XMLCite \textit{B. Hannebicque} and \textit{É. Herbin}, Stochastic Processes Appl. 154, 154--196 (2022; Zbl 1510.60042) Full Text: DOI References: [1] Abraham, Céline; Le Gall, Jean-François, Excursion theory for Brownian motion indexed by the Brownian tree, J. Eur. Math. Soc. 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