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Abstract
We study the convergence in probability in the nonstandard 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 M1convergence, prove a criterion for the M1convergence 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 OrnsteinUhlenbeck processes with diagonalisable generators.
Original language  English 

Pages (fromto)  271305 
Number of pages  35 
Journal  STOCHASTIC ANALYSIS AND APPLICATIONS 
Volume  33 
Issue number  2 
Early online date  2 Feb 2015 
DOIs  
Publication status  Published  2015 
Keywords
 M1 Skorokhod topology
 stochastic convolution integral
 Lévy process
 Hilbert space
 Banach space
 convergence in probability
 OrnsteinUhlenbeck process
 integrated OrnsteinUhlenbeck process
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 1 Finished

Cylindrical Levy processes and their applications
EPSRC Engineering and Physical Sciences Research Council
1/06/2012 → 31/05/2014
Project: Research