Stochastic B-series analysis of iterated Taylor methods

TL;DR

This paper develops a stochastic B-series framework to analyze the convergence of iterated Taylor methods for stochastic differential equations, considering various iterative schemes, predictors, and iteration counts, with theoretical and numerical validation.

Contribution

It introduces a stochastic B-series approach to analyze convergence of iterated Taylor methods for SDEs, encompassing different iteration strategies and convergence types.

Findings

Convergence results depend on Taylor order, iteration method, predictor, and iterations.

Theoretical analysis covers Itô and Stratonovich SDEs, weak and strong convergence.

Numerical experiments support the theoretical findings.

Abstract

For stochastic implicit Taylor methods that use an iterative scheme to compute their numerical solution, stochastic B--series and corresponding growth functions are constructed. From these, convergence results based on the order of the underlying Taylor method, the choice of the iteration method, the predictor and the number of iterations, for It\^o and Stratonovich SDEs, and for weak as well as strong convergence are derived. As special case, also the application of Taylor methods to ODEs is considered. The theory is supported by numerical experiments.