Non-standard Skorokhod convergence of Levy-driven convolution integrals in Hilbert spaces

Ilya Pavlyukevich, Markus Riedle

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We study the convergence in probability in the non-standard M1 Skorokhod topology of the Hilbert valued stochastic convolution integrals of the type ∫Fγ(t−s)dL(s) to a process ∫F(t−s)dL(s) driven by a Lévy process L. In Banach spaces we introduce strong, weak and product modes of M1-convergence, prove a criterion for the M1-convergence in probability of stochastically continuous càdlàg processes in terms of the convergence in probability of the finite dimensional marginals and a good behaviour of the corresponding oscillation functions, and establish criteria for the convergence in probability of Lévy driven stochastic convolutions. The theory is applied to the infinitely dimensional integrated Ornstein-Uhlenbeck processes with diagonalisable generators.
Original languageEnglish
Pages (from-to)271-305
Number of pages35
Issue number2
Early online date2 Feb 2015
Publication statusPublished - 2015


  • M1 Skorokhod topology
  • stochastic convolution integral
  • Lévy process
  • Hilbert space
  • Banach space
  • convergence in probability
  • Ornstein-Uhlenbeck process
  • integrated Ornstein-Uhlenbeck process


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