Cubature Methods For Stochastic (Partial) Differential Equations In Weighted Spaces

TL;DR

This paper develops high-order cubature methods for stochastic PDEs in weighted spaces, achieving optimal convergence rates even with unbounded payoffs and characteristics, by leveraging a functional analytic framework and symmetry assumptions.

Contribution

It introduces a novel weighted space framework for SPDEs, establishing optimal convergence rates for cubature methods under new symmetry and stability conditions.

Findings

Established high-order weak approximation schemes for SPDEs in weighted spaces.

Proved stability of local approximation operators in the infinite-dimensional setting.

Achieved optimal convergence rates for nonsmooth payoffs with exponential growth.

Abstract

The cubature on Wiener space method, a high-order weak approximation scheme, is established for SPDEs in the case of unbounded characteristics and unbounded payoffs. We first introduce a recently described flexible functional analytic framework, so called weighted spaces, where Feller-like properties hold. A refined analysis of vector fields on weighted spaces then yields optimal convergence rates of cubature methods for stochastic partial differential equations of Da Prato-Zabczyk type. The ubiquitous stability for the local approximation operator within the functional analytic setting is proved for SPDEs, however, in the infinite dimensional case we need a newly introduced assumption on weak symmetry of the cubature formula. In finite dimensions, we use the UFG condition to obtain optimal rates of convergence on non-uniform meshes for nonsmooth payoffs with exponential growth.