Composition of stochastic B-series with applications to implicit Taylor methods

TL;DR

This paper develops a new representation formula for stochastic B-series, enabling the derivation of order conditions for implicit Taylor methods, and introduces a family of strong order 1.5 methods for Itô SDEs.

Contribution

It provides the first order conditions for implicit Taylor methods using rooted trees and constructs a family of strong order 1.5 methods for Itô SDEs.

Findings

Derived order conditions for implicit Taylor methods

Constructed a family of strong order 1.5 Taylor methods

Applied the conditions to develop new numerical schemes

Abstract

In this article, we construct a representation formula for stochastic B-series evaluated in a B-series. This formula is used to give for the first time the order conditions of implicit Taylor methods in terms of rooted trees. Finally, as an example we apply these order conditions to derive in a simple manner a family of strong order 1.5 Taylor methods applicable to It\^o SDEs.