Approximation and convergence of solutions to semilinear stochastic evolution equations with jumps

TL;DR

This paper establishes the continuity and convergence of solutions to semilinear stochastic evolution equations with jumps, driven by martingales or Poisson measures, under specific operator and coefficient conditions.

Contribution

It provides new results on the joint continuity and convergence of solutions to these equations, extending understanding to cases with jumps and operator convergence.

Findings

Solutions are jointly continuous with respect to initial data and coefficients.

Under certain operator convergence conditions, solutions to the equations converge.

The results apply to equations driven by martingales or Poisson random measures.

Abstract

We prove that the mild solution to a semilinear stochastic evolution equation on a Hilbert space, driven by either a square integrable martingale or a Poisson random measure, is (jointly) continuous, in a suitable topology, with respect to the initial datum and all coefficients. In particular, if the leading linear operators are maximal (quasi-)monotone and converge in the strong resolvent sense, the drift and diffusion coefficients are uniformly Lipschitz continuous and converge pointwise, and the initial data converge, then the solutions converge.