B–Series Analysis of Stochastic Runge–Kutta Methods That Use an Iterative Scheme to Compute Their Internal Stage Values

Abstract

In recent years, implicit stochastic Runge-Kutta (SRK) methods have been developed both for strong and weak approximations. For these methods, the stage values are only given implicitly. However, in practice these implicit equations are solved by iterative schemes such as simple iteration, modified Newton iteration or full Newton iteration. We employ a unifying approach for the construction of stochastic B-series which is valid both for Itô-and Stratonovich-stochastic differential equations (SDEs) and applicable both for weak and strong convergence to analyze the order of the iterated Runge-Kutta method. Moreover, the analytical techniques applied in this paper can be of use in many other similar contexts.

General order conditions for stochastic partitioned Runge–Kutta methods

“…B-series for SDEs were developed by Burrage and Burrage [4,5,7] for strong convergence of Stratonovich SDEs, by Komori, Mitsui and Sugiura [17] and Komori [16] for weak convergence of Stratonovich SDEs, and by Rößler [25,26] for weak convergence in both the Itô and Stratonovich case. A unified theory for B-series encompassing both weak and strong convergence for both Itô and Stratonovich SDEs was given in [9]. In the following we will generalize this to SPRKs.…”

A Runge-Kutta method for index 1 stochastic differential-algebraic equations with scalar noise

“…Now, by applying the colored rooted tree theory for Itô SDEs given in [4,13,14], order conditions for the coefficients of the SRK method (2.3) can be easily calculated, making use of the vector e = (1, . .…”

Stochastic B-Series and Order Conditions for Exponential Integrators

“…Proof. The proof of this lemma is given in [6]. ⊓ ⊔ Applying Lemma 5 to the functions g m on the right hand side of (1) gives…”

“…Naturally then, such series have also been derived for stochastic differential equations (SDEs) by several authors, see e.g. [6] for an overview.…”