The Wiener integral with respect to second order processes with stationary increments

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Abstract

We prove that for any second order stochastic process X with stationary increments with continuous paths and continuous variance function, there exists a tempered measure μ (for which we give an explicit expression) related with the domain of the Wiener integral with respect to X as follows: the space of tempered distributions f such that the Fourier transform of f is square integrable with respect to μ is always a dense subset of the domain of the Wiener integral. Moreover, we provide sufficient conditions on μ in order that the domain of the integral is exactly this space of distributions. We apply our results to the fractional Brownian motion. In particular, it is proved that the domain of the Wiener integral with respect to the fractional Brownian motion with Hurst parameter H > 1 / 2 contains distributions that are not given by locally integrable functions, this fact was suggested by Pipiras and Taqqu (2000) in [5]. We have also considered the example of the process given by Ornstein and Uhlenbeck as a model for the position of a Brownian particle. © 2010 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)607-620
JournalJournal of Mathematical Analysis and Applications
Volume366
DOIs
Publication statusPublished - 15 Jun 2010

Keywords

  • Random distributions
  • Second order process with stationary increments
  • Wiener integral

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