SPDE bridges with observation noise and their spatial approximation

TL;DR

This paper develops a framework for SPDE bridges with observation noise, analyzing their spatial discretizations and providing convergence rates, which are crucial for accurate numerical simulations of stochastic processes in Hilbert spaces.

Contribution

It introduces a novel approach to SPDE bridges with observation noise and derives explicit convergence rates for spectral and finite element discretizations.

Findings

Convergence rates are established for the discretizations.

Rates match those of the original SPDE under rough noise.

A general framework for spatial discretization is proposed.

Abstract

This paper introduces SPDE bridges with observation noise and contains an analysis of their spatially semidiscrete approximations. The SPDEs are considered in the form of mild solutions in an abstract Hilbert space framework suitable for parabolic equations. They are assumed to be linear with additive noise in the form of a cylindrical Wiener process. The observational noise is also cylindrical and SPDE bridges are formulated via conditional distributions of Gaussian random variables in Hilbert spaces. A general framework for the spatial discretization of these bridge processes is introduced. Explicit convergence rates are derived for a spectral and a finite element based method. It is shown that for sufficiently rough observation noise, the rates are essentially the same as those of the corresponding discretization of the original SPDE.