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

TL;DR

This paper investigates the convergence of Levy-driven stochastic convolution integrals in Hilbert spaces under the non-standard M_1 Skorokhod topology, providing criteria for convergence and applying the results to infinite-dimensional Ornstein-Uhlenbeck processes.

Contribution

It introduces new modes of M_1 convergence in Banach spaces, establishes criteria for convergence of Levy-driven convolutions, and applies the theory to infinite-dimensional Ornstein-Uhlenbeck processes.

Findings

Established criteria for M_1 convergence in probability of stochastic convolutions.

Proved convergence in probability based on finite-dimensional marginals and oscillation functions.

Applied the theory to infinite-dimensional integrated Ornstein-Uhlenbeck processes.

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.