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

Stochastic Runge-Kutta Methods with Deterministic High Order for Ordinary Differential Equations

Our aim is to show that the embedding of deterministic Runge-Kutta methods with higher order than necessary order to achieve a weak order can enrich the properties of stochastic Runge-Kutta methods with respect to not only practical errors but also stability. This will be done through the comparisons between our new schemes and an efficient weak second order scheme with minimized error constant proposed by Debrabant and Robler (2009)

Stochastic Runge-Kutta methods with deterministic high order for ordinary differential equations

stochastic Runge-Kutta (SRK) methods for non-commutative stochastic differential equations (SDEs). As a result, we have obtained weak second order SRK methods which have good properties with respect to not only practical errors but also mean square stability. In our stability analysis, as well as a scalar test equation with complex-valued parameters, we have used a multi-dimensional non-commutative test SDE. The performance of our new schemes will be shown through comparisons with an efficient and optimal weak second order scheme proposed by Debrabant and Rößler (Appl. Numer. Math. 59:582–594, 2009)

A Stochastic Method for All Seasons

It is well known that the numerical solution of stochastic ordinary differential equations leads to a step size reduction when explicit methods are used. This has led to a plethora of implicit methods with a wide variety of stability properties. However, for stochastic problems whose eigenvalues lie near the negative real axis, explicit methods with extended stability regions can be very effective. In this paper we extend these ideas to the stochastic realm and present a family of weak order two explicit stochastic Runge-Kutta methods with extended stability intervals that can be used to solve a variety of non-stiff and stiff problems