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 $\int_0^t F_\gamma(t-s)\,d L(s)$ to a process $\int_0^t F(t-s)\, d L(s)$ driven by a L\'evy 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 c\`adl\`ag 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 L\'evy driven stochastic convolutions. The theory is applied to the infinitely dimensional integrated Ornstein--Uhlenbeck processes with diagonalisable generators.