Runge-Kutta Methods for the Numerical Solution of Stochastic Differential Equations

The aim of this thesis is to provide an easy approach to the calculation of stochastic Runge-Kutta schemes, also for higher order of convergence. Therefore, Runge-Kutta methods for the numerical solution of stochastic differential equations (SDEs) converging in the weak sense are considered, and coloured rooted trees composed of three different types of nodes are introduced. Under the application of coloured trees, the Taylor expansion of the expectation of some functional of the exact solution together with an estimation of the truncation error for both, Ito and Stratonovich SDEs w.r.t. a one-dimensional Wiener process is proved. This is followed by the introduction of a very general dass of stochastic Runge-Kutta methods permitting a wide choice for the random variables used by the method. Using the introduced rooted trees, the Taylor expansion of the expectation of a functional of the approximation process is also proved.

The main theorem introduced in this study gives general conditions for the coefficients of a stochastic Runge-Kutta method such that it converges with a desired order p in the weak sense w.r.t. both Ito and Stratonovich SDEs. The main advantage of the presented conditions is that they can be calculated very easily without taking care of any derivatives. This is due to the introduced coloured rooted tree theory.

The idea of adding variable indices to the stochastic nodes of the considered coloured rooted trees leads to similar results in the case of SDEs w.r.t. a multidimensional Wiener process. Therefore, the considered dass of stochastic Runge­ Kutta methods is generalized and a generalization of the main theorem taking into account the variable indices is stated. Also, conditions for the coefficients of a Runge-Kutta method converging weakly with some order p can be calculated easily.

Finally, conditions for the coefficients for some explicit stochastic Runge-Kutta methods of weak order 2.0 are calculated. Some coefficients for schemes approximating both Itö SDEs and Stratonovich SDEs are presented. A simulation study reveals the good performance of the proposed schemes in comparison to wellknown schemes.