Runge-Kutta methods for rough differential equations

We study Runge-Kutta methods for rough differential equations which can be used to calculate solutions to stochastic differential equations driven by processes that are rougher than a Brownian motion. We use a Taylor series representation (B-series) for both the numerical scheme and the solution of the rough differential equation in order to determine conditions that guarantee the desired order of the local error for the underlying Runge-Kutta method. Subsequently, we prove the order of the global error given the local rate. In addition, we simplify the numerical approximation by introducing a Runge-Kutta scheme that is based on the increments of the driver of the rough differential equation. This simplified method can be easily implemented and is computational cheap since it is derivative-free. We provide a full characterization of this implementable Runge-Kutta method meaning that we provide necessary and sufficient algebraic conditions for an optimal order of convergence in case that the driver, e.g., is a fractional Brownian motion with Hurst index $\frac{1}{4} < H \leq \frac{1}{2}$. We conclude this paper by conducting numerical experiments verifying the theoretical rate of convergence

Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs

We introduce a new algebraic framework based on a modification (called exotic) of aromatic Butcher-series for the systematic study of the accuracy of numerical integrators for the invariant measure of a class of ergodic stochastic differential equations (SDEs) with additive noise. The proposed analysis covers Runge-Kutta type schemes including the cases of partitioned methods and postprocessed methods. We also show that the introduced exotic aromatic B-series satisfy an isometric equivariance property.Comment: 33 page

General order conditions for stochastic partitioned Runge-Kutta methods

In this paper stochastic partitioned Runge-Kutta (SPRK) methods are considered. A general order theory for SPRK methods based on stochastic B-series and multicolored, multishaped rooted trees is developed. The theory is applied to prove the order of some known methods, and it is shown how the number of order conditions can be reduced in some special cases, especially that the conditions for preserving quadratic invariants can be used as simplifying assumptions

Runge-Kutta methods for third order weak approximation of SDEs with multidimensional additive noise

A new class of third order Runge-Kutta methods for stochastic differential equations with additive noise is introduced. In contrast to Platen's method, which to the knowledge of the author has been up to now the only known third order Runge-Kutta scheme for weak approximation, the new class of methods affords less random variable evaluations and is also applicable to SDEs with multidimensional noise. Order conditions up to order three are calculated and coefficients of a four stage third order method are given. This method has deterministic order four and minimized error constants, and needs in addition less function evaluations than the method of Platen. Applied to some examples, the new method is compared numerically with Platen's method and some well known second order methods and yields very promising results.Comment: Two further examples added, small correction

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

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.Comment: slight changes to improve readability. Changes resulting from the publishing process may not be reflected in the preprint versio

Stochastic B-series analysis of iterated Taylor methods

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