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

Abstract

We study the convergence in probability in the non-standard $M_1$ Skorokhod topology of the Hilbert valued stochastic convolution integrals of the type ${\int }_0^t{F}_{\gamma }(t-s)dL(s)$ to a process ${\int }_0^{t}F(t-s)dL(s)$ driven by a Lévy process $L$. In Banach spaces we introduce strong, weak and product modes of $M_1$-convergence, prove a criterion for the $M_1$-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.

34 pages, 1 figure

References in corpus (3)

• Small noise asymptotics and first passage times of integrated Ornstein-Uhlenbeck processes driven by $\alpha$-stable Lévy processes
• Ornstein-Uhlenbeck processes driven by cylindrical Lévy processes
• Structure of infinitely divisible semimartingales