A Semigroup Point Of View On Splitting Schemes For Stochastic (Partial) Differential Equations

TL;DR

This paper develops a semigroup framework for analyzing splitting schemes in stochastic (partial) differential equations, providing new insights into convergence rates for equations with unbounded growth.

Contribution

It introduces normed spaces that ensure strong continuity of semigroups of positive operators, enabling optimal convergence analysis for splitting schemes in infinite-dimensional stochastic PDEs.

Findings

Established strong continuity of semigroups on controlled growth spaces.

Derived optimal convergence rates for splitting schemes.

Applied results to Da Prato-Zabczyk and HJM interest rate models.

Abstract

We construct normed spaces of real-valued functions with controlled growth on possibly infinite-dimensional state spaces such that semigroups of positive, bounded operators $(P_t)_{t\ge 0}$ thereon with $\lim_{t\to 0+}P_t f(x)=f(x)$ are in fact strongly continuous. This result applies to prove optimal rates of convergence of splitting schemes for stochastic (partial) differential equations with linearly growing characteristics and for sets of functions with controlled growth. Applications are general Da Prato-Zabczyk type equations and the HJM equations from interest rate theory.