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

TY - JOUR

T1 - Non-standard Skorokhod convergence of Levy-driven convolution integrals in Hilbert spaces

AU - Pavlyukevich, Ilya

AU - Riedle, Markus

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

KW - M1 Skorokhod topology

KW - stochastic convolution integral

KW - Lévy process

KW - Hilbert space

KW - Banach space

KW - convergence in probability

KW - Ornstein-Uhlenbeck process

KW - integrated Ornstein-Uhlenbeck process

U2 - 10.1080/07362994.2014.988358

DO - 10.1080/07362994.2014.988358

M3 - Article

SN - 0736-2994

VL - 33

SP - 271

EP - 305

JO - STOCHASTIC ANALYSIS AND APPLICATIONS

JF - STOCHASTIC ANALYSIS AND APPLICATIONS

IS - 2

ER -