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.     

Continuous weak approximation for stochastic differential equations

A convergence theorem for the continuous weak approximation of the solution of stochastic differential equations (SDEs) by general one-step methods is proved, which is an extension of a theorem due to Milstein. As an application, uniform second order conditions for a class of continuous stochastic Runge–Kutta methods containing the continuous extension of the second order stochastic Runge–Kutta scheme due to Platen are derived. Further, some coefficients for optimal continuous schemes applicable to Itô SDEs with respect to a multi–dimensional Wiener process are presented.     

Runge–Kutta Lawson schemes for stochastic differential equations

In this paper, we present a framework to construct general stochastic Runge–Kutta Lawson schemes. We prove that the schemes inherit the consistency and convergence properties of the underlying Runge–Kutta scheme, and confirm this in some numerical experiments. We also investigate the stability properties of the methods and show for some examples, that the new schemes have improved stability properties compared to the underlying schemes.

     

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 Itô 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.     

Diagonally drift-implicit Runge–Kutta methods of weak order one and two for Itô SDEs and stability analysis

The class of stochastic Runge–Kutta methods for stochastic differential equations due to Rößler is considered. Coefficient families of diagonally drift-implicit stochastic Runge–Kutta (DDISRK) methods of weak order one and two are calculated. Their asymptotic stability as well as mean-square stability (MS-stability) properties are studied for a linear stochastic test equation with multiplicative noise. The stability functions for the DDISRK methods are determined and their domains of stability are compared to the corresponding domain of stability of the considered test equation. Stability regions are presented for various coefficients of the families of DDISRK methods in order to determine step size restrictions such that the numerical approximation reproduces the characteristics of the solution process.     

Families of efficient second order Runge–Kutta methods for the weak approximation of Itô stochastic differential equations

Recently, a new class of second order Runge–Kutta methods for Itô stochastic differential equations with a multidimensional Wiener process was introduced by Rößler [A. Rößler, Second order Runge–Kutta methods for Itô stochastic differential equations, Preprint No. 2479, TU Darmstadt, 2006]. In contrast to second order methods earlier proposed by other authors, this class has the advantage that the number of function evaluations depends only linearly on the number of Wiener processes and not quadratically. In this paper, we give a full classification of the coefficients of all explicit methods with minimal stage number. Based on this classification, we calculate the coefficients of an extension with minimized error constant of the well-known RK32 method [J.C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, West Sussex, 2003] to the stochastic case. For three examples, this method is compared numerically with known order two methods and yields very promising results.     

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 Itô SDEs.