Strong convergence of some Magnus-type schemes for the finite element discretization of non-autonomous parabolic SPDEs driven by additive fractional Brownian motion and Poisson random measure

TL;DR

This paper establishes strong convergence results for novel numerical schemes approximating non-autonomous semilinear SPDEs driven by both fractional Brownian motion and Poisson measures, using finite element spatial discretization and Magnus-type or Euler time integrators.

Contribution

It introduces the first numerical methods for non-autonomous semilinear SPDEs driven by both fractional Brownian motion and Poisson measures, analyzing their convergence properties.

Findings

Convergence orders depend on initial data regularity and noise characteristics.

Finite element method effectively discretizes space for these SPDEs.

Magnus-type and Euler schemes achieve strong convergence in mean-square sense.

Abstract

The aim of this work is to provide the strong convergence results of numerical approximations of a general second order non-autonomous semilinear stochastic partial differential equation (SPDE) driven simultaneously by an additive fractional Brownian motion (fBm) with Hurst parameter H \in (1/2,1) and a Poisson random measure, more realistic in modelling real world phenomena. Approximations in space are performed by the standard finite element method and in time by the stochastic Magnus-type integrator or the linear semi-implicit Euler method. We investigate the mean-square errors estimates of our fully discrete schemes and the results show how the convergence orders depend on the regularity of the initial data and the driven processes. To the best of our knowledge, these two schemes are the first numerical methods to approximate the non-autonomous semilinear stochastic partial differential equation (SPDE) driven simultaneously by an additive fractional Brownian motion with Hurst parameter H and a Poisson random measure.