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

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

2012-01-19

arxiv.org/abs/1201.4024v1

Mathematics

Probability

Scientific paper

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.

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