Discretizations of Stochastic Evolution Equations in Variational Approach Driven by Jump-Diffusion

TL;DR

This paper develops and analyzes finite element and Galerkin discretizations for stochastic evolution equations driven by jump-diffusion noise, proving their convergence under certain growth conditions.

Contribution

It introduces explicit and implicit discretization schemes for jump-diffusion stochastic equations within a variational framework and establishes their convergence.

Findings

Explicit and implicit schemes are convergent.

Convergence proofs are provided under polynomial and linear growth conditions.

Finite element and Galerkin methods are effectively applied to jump-diffusion equations.

Abstract

Stochastic evolution equations with compensated Poisson noise are considered in the variational approach with monotone and coercive coefficients. Here the Poisson noise is assumed to be time-homogeneous with $\sigma$-finite intensity measure on a metric space. By using finite element methods and Galerkin approximations, some explicit and implicit discretizations for this equation are presented and their convergence is proved. Polynomial growth condition and linear growth condition are assumed on the drift operator, respectively for the implicit and explicit schemes.